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k. CONFEl J d 

CIVIL ENGINEERING STUDIES 

SANITARY ENGINEERING SERIES NO. 48 




ENGINEERING LIBRARY 




URBANA, ILLINOIS 61801 



CON! 

A PROBABILISTIC ANALYSIS 

OF DISSOLVED OXYGEN-BIOCHEMICAL OXYGEN 

DEMAND RELATIONSHIP IN STREAMS 



ROOM 



By 
VEERASAMY KOTHANDARAAAAN 

Supported by 

OFFICE OF WATER RESOURCES RESEARCH 

U.S. DEPARTMENT OF THE INTERIOR 

RESEARCH PROJECT B-006-ILL 



DEPARTMENT OF CIVIL ENGINEERING 

UNIVERSITY OF ILLINOIS 

URBANA, ILLINOIS 

JUNE, 1968 



A PROBABILISTIC ANALYSIS OF DISSOLVED OXYGEN- 
BIOCHEMICAL OXYGEN DEMAND RELATIONSHIP IN STREAMS 

Veerasamy Kothandaraman, Ph.D. 

Department of Civil Engineering 
University of Illinois, (1968) 



Until recently, the classical Streeter-P helps equation was 
widely used in predicting the dissolved oxygen deficit to organic 
waste loads discharged into a stream. However, the wide deviation 
of predicted values of oxygen deficit from the observed value lead 
several investigators to modify the S tree ter -Phelps formulation by 
taking into account other factors like benthal demand, photosynthesis, 
algal respiration, etc., which affect the dissolved oxygen in the 
stream, in order to narrow the gap between the observed and the pre- 
dicted values. Other investigators postulated statistical models 
for predicting critical oxygen concentrations using stream flow, 
temperature, and 5-day BOD as independent variables, mainly to avoid 
the errors involved in the predictions due to the fluctuations in 
the values of deoxygenation and reaeration coefficients. 

This work attempts to predict dissolved oxygen deficits in a 
stream with known initial conditions by taking into account the varia- 
tions in deoxygenation and reaeration coefficients. A hypothetical 
stream situation is used to establish the significance in predicting 
dissolved oxygen deficit. Statistical models are formulated and 
tested for the variations in these coefficients using published data. 
Simulation techniques using the Monte Carlo method are employed in 



rO. ENGI \ LIBRARY z 

predicting the probabilistic variation in dissolved oxygen deficits 
for known initial conditions and the results are verified with the 
survey data observed for the Ohio River-Cincinnati Pool reach. The 
predicted results using probabilistic model are found to agree with 
the observed values within practical limits and give more consistent 
results than conventional methods. 



iii 



ACKNOWLEDGMENTS 

The author wishes to express his sincere appreciation to all 
those whose counsel and assistance aided in the successful completion 
of this work. He especially wishes to thank the following: 

Professor B. B. Ewing for his valuable guidance, counsel, 
and encouragement and for the many hours he spent in reading and dis- 
cussing the thesis. 

Professor A. H. S. Ang for the valuable suggestions he gave 
in the early stages of this study and for the time he spent in reviewing 
the work at different stages of its development. 

Other members of his academic committee for their advice 
and general comments that have considerably improved the final thesis. 

The staff of the University of Illinois Digital Computer 
Laboratory for the facilities made available to him in the use of the 
I.B.M. 7094-1407 system (partially supported by a grant from the 
National Science Foundation, NSFGP 700), and the staff of the University 
of Illinois Statistical Service, Research Unit for their help in 
debugging computer programs. 

His wife and daughter for their cooperation and understanding. 

This study was supported by a matching-fund grant No. B-006-ILL 
from the Office of Water Resources Research, United States Department 
of the Interior, as authorized under the Water Resources Research Act 
of 1964. 



iv 



TABLE OF CONTENTS 

Page 

ACKNOWLEDGMENTS iii 

LIST OF TABLES vii 

LIST OF FIGURES viii 

I. INTRODUCTION 1 

OBJECTIVE OF THE STUDY 4 

SCOPE OF THE STUDY 6 

II. LITERATURE REIVEW 7 

III. SENSITIVITY ANALYSIS 24 

DEFINITION 24 

PROCEDURE 24 

Nominal Values in the Streeter-P helps Equation. • • 25 

Temperature Effects on Reaction Coefficients. ... 27 

RESULTS 29 

IV. PROBABILISTIC VARIATIONS IN VELOCITY COEFFICIENTS ... 43 

VARIATIONS IN THE DEOXYGENATION COEFFICIENT 43 

Kj Values of the Ohio River Samples ........ 45 

Randomness Test 51 

Determination of Sequence Order ........ 52 

Runs Up and Down Test 53 

Turning Points Test 55 

Hypothesis and Hypothesis Testing ......... 57 

Chi -Square Test ......... 60 

Kolmogorov-Smirnov Test 61 



V 



TABLE OF CONTENTS (Continued) 

Page 

VARIATIONS IN REAERATION COEFFICIENT 63 

Background ••••••••• 63 

Modified Regression Equation .•••. 68 

V. MONTE CARLO METHOD AND SIMULATION MODEL FOR STREAM 

ASSIMILATIVE CAPACITIES . . 74 

MONTE CARLO METHOD 74 

General 74 

Random Sampling from Specified Distribution .... 77 

Sources of Random Numbers •••• 77 

Random Number Generators • 78 

Development of Pseudo-Random Numbers 78 

Multiplicative Congruence Method 79 

Transformation of Random Numbers. ... 80 

Variance Reducing Techniques. •••.••...•• 82 

Straightforward Sampling 82 

Estimation of Sample Sise ...... 84 

SIMULATION IN WATER RESOURCES SYSTEMS 85 

MODEL FOR STREAM ASSIMILATIVE CAPACITY 88 

Factors Affecting DO and BOD. 88 

Equations for BOD and DO Profiles 92 

Application to the Problem under Study • 93 

VI. APPLICATION OF MONTE CARLO METHOD TO THE OHIO RIVER 

SURVEY DATA 96 

CHOICE OF PUBLISHED RIVER SURVEY DATA 96 



vi 



TABLE OF CONTENTS (Continued) 

Page 

VALUES OF THE PARAMETERS USED IN THE MODEL. 97 

Deoxygenation Coefficient ........ 97 

Effect of Sludge Deposits 98 

Reaeration Coefficient 100 

Photosynthesis and Respiration. .......... 103 

SAMPLE SIZE IN SIMULATION STUDIES OF THE OHIO RIVER . . 103 

RESULTS OF THE MONTE CARLO METHOD APPLIED TO THE OHIO 

RIVER 104 

VII. DISCUSSION 114 

VIII. ENGINEERING SIGNIFICANCE 122 

IX. SUMMARY, CONCLUSIONS, AND SUGGESTIONS FOR FUTURE STUDY. 124 

SUMMARY OF THE STUDY 124 

CONCLUSIONS 126 

SUGGESTIONS FOR FUTURE STUDY 127 

REFERENCES 129 

APPENDIX A NOTATION 136 

APPENDIX B FLOW DIAGRAM FOR OXYGEN SAG COMPUTATIONS; 

SENSITIVITY ANALYSIS 140 

APPENDIX C FLOW DIAGRAM FOR SOLVING PROBABILISTIC MODEL 

FOR DO-BOD RELATIONSHIP 142 

APPENDIX D BOD PROGRESSION DATA FOR THE OHIO RIVER 

SAMPLES (1957 SURVEY) 145 

APPENDIX E HYDRAULIC CHARACTERISTICS OF REACHES IN 

TENNESSEE RIVERS USED IN MULTIPLE CORRELATION 

ANALYSIS 155 



vii 



LIST OF TABLES 



Table Page 

1 NOMINAL VALUES OF REACTION COEFFICIENTS Kj 

AND K 2 29 

2 EFFECT OF VARIATION IN K« ON PERCENT ERROR IN 
PREDICTING CRITICAL DO AT 10<>C 32 

3 EFFECT OF VARIATION IN K 2 ON PERCENT ERROR IN 
PREDICTING CRITICAL DO AT 20°C 33 

4 EFFECT OF VARIATION IN K 2 ON PERCENT ERROR IN 
PREDICTING CRITICAL DO AT 30°C. 34 

5 EFFECT OF VARIATION IN K t ON PERCENT ERROR IN 
PREDICTING CRITICAL DO AT DIFFERENT TEMPERATURES. 37 

6 DISTRIBUTION OF RUNS UP AND DOWN FOR THE SEQUENCE 
OF OBSERVATIONS FOR Kj VALUES OF THE OHIO RIVER 
SAMPLES 55 



DISTRIBUTION OF "PHASES" FOR THE SEQUENCE OF 
OBSERVATIONS FOR K x VALUES OF THE OHIO RIVER 
SAMPLES 56 



8 FREQUENCY DISTRIBUTION OF OBSERVED K x VALUES OF 

THE OHIO RIVER SAMPLES 57 

9 CHI -SQUARE TEST FOR THE GOODNESS OF FIT CON- 
CERNING THE HYPOTHESIS FOR OBSERVED K x VALUES . . 61 

10 KOLMOGOROV-SMIRNOV TEST FOR THE GOODNESS OF FIT 
CONCERNING THE HYPOTHESIS FOR OBSERVED J^ VALUES. 62 

11 VALUES OF PARAMETERS USED IN THE SIMULATION MODEL 102 

12 INITIAL AND FINAL CONDITIONS OF THE OHIO RIVER 
SAMPLES FOR SIX DIFFERENT CASES USED TO VERIFY 

MONTE CARLO PREDICTIONS 107 

13 FREQUENCY DISTRIBUTION OF PROBABLE VALUES OF DO 
DEFICITS 108 

14 ACTUAL AND PREDICTED DO DEFICITS 113 



vlli 



LIST OF FIGURES 

Figure Page 

1 DISSOLVED OXYGEN SAG CURVE 12 

2 EFFECT OF VARIATIONS INK, VALUES ON PERCENT ERROR 
IN PREDICTING THE CRITICAL DO AT DIFFERENT TEMPER- 
ATURES 35 

3 EFFECT OF VARIATIONS IN K ± VALUES ON PERCENT ERROR 

IN PREDICTING CRITICAL DO AT DIFFERENT TEMPERATURES 38 

4 COMPARISON OF THE EFFECTS OF VARIATIONS IN K, 
AND K 2 ON PERCENT ERROR IN PREDICTING THE CRITI- 
CAL DO AT 10°C 39 

5 COMPARISON OF THE EFFECTS OF VARIATIONS IN K. 
AND K 2 ON PERCENT ERROR IN PREDICTING THE CRITI- 
CAL DO AT 20°C 40 



COMPARISON OF THE EFFECTS OF VARIATIONS IN K x 
AND K 2 ON PERCENT ERROR IN PREDICTING THE CRITI- 
CAL DO AT 30°C 41 



7 FLOW DIAGRAM FOR COMPUTING K x AND ULTIMATE FIRST 
STAGE BOD VALUES USING REED-THERIAULT METHOD. . . 48 

8 BOD PROGRESSION OF TREATMENT PLANT EFFLUENTS. . . 49 

9 BOD PROGRESSIONS OF RIVER SAMPLES 50 

10 FREQUENCY PLOTS FOR K, VALUES OF THE OHIO RIVER 
SAMPLES 58 

11 CUMULATIVE PROBABILITY DISTRIBUTION OF Kj VALUES 

FOR THE OHIO RIVER SAMPLES. 59 

12 OBSERVED VERSUS PREDICTED VALUES OF K 2 71 

13 PROBABILITY DISTRIBUTION OF PERCENT RESIDUAL ERRORS 

IN PREDICTING K 2 VALUES 72 

14 FREQUENCY DISTRIBUTION OF DO DEFICIT IN CASE 

STUDY NO. 1 110 

15 FREQUENCY DISTRIBUTION OF DO DEFICIT IN CASE 

STUDY NO. 2 110 



IX 



LIST OF FIGURES (Continued) 



Figure Page 

16 FREQUENCY DISTRIBUTION OF DO DEFICIT IN CASE 

STUDY NO. 3 Ill 

17 FREQUENCY DISTRIBUTION OF DO DEFICIT IN CASE 

STUDY NO. 4 , r Ill 

18 FREQUENCY DISTRIBUTION OF DO DEFICIT IN CASE 

STUDY NO. 5 112 

19 FREQUENCY DISTRIBUTION OF DO DEFICIT IN CASE 

STUDY NO. 6 112 

20 RANGE OF DO AND THE ASSOCIATED PROBABILITY AT 
DIFFERENT SECTIONS OF THE OHIO RIVER FOR THE 

CASE STUDY NO. 5 119 



I. INTRODUCTION 

The rapid expansion of population and industrialization 
has resulted in increasingly difficult problems of water resources 
management. The most critical of these problems is the protection 
of water resources from the ravages of pollution by the discharge of 
wastes which are increasing in both volume and complexity. There is a 
general awakening and a demand that something must be done about stream 
pollution. The daily news media report various aspects of water pol- 
lution with considerable frequency and in the past decade many national 
laws have been enacted dealing with Federal participation in pollution 
control activities. 

If a pollution control program is to be both effective and 
economical, engineers must possess the ability to predict the effects 
of specific waste discharges on the environment. Without this know- 
ledge, administrative determination of the required degree of treatment 
for waste discharges, existing or proposed, can only be speculative. 
A low estimate of the required degree of treatment will result in un- 
desirable conditions, while an overestimate will create unjustifiable 
economic burdens on the waste discharger without commensurate benefit 
to the environment. 

The determination of acceptable levels of water quality is 
usually the task of a regulatory agency. These governmental agencies 
have used several approaches to control the water quality in streams. 
One approach is through the use of treatment standards which require 
specified degrees of treatment, usually in terms of biochemical oxygen 
demand (BOD) reduction, suspended solids and coliform removal. Effluent 



standards have also been established wlich limit the concentrations 
of the various constituents in the effluent released into the stream. 
These regulatory activities are attuned to the concept of equity, which 
require the same degree of removal of BOD and solids from all waste 
sources irrespective of the volume of waste (Jacobs, 1965). These 
standards are used because they are relatively easier to administer, 
but each one deals only indirectly with the basic problem of stream 
quality. 

The establishment of stream standards requires the polluter 
to regulate his effluent such that at least a minimum level of stream 
quality is maintained at all times* Though the aim of all the three 
types of standards is to assure acceptable levels of stream quality, it 
is directly accomplished only in the case of stream standards. The 
adoption of stream standards as a tool for regulatory control of pollu- 
tion is not without limitations. In the first place it is very difficult 
to administer and enforce, especially if there is more than one polluter. 
Also, the application of rigid stream standards to large areas may well 
become a barrier to orderly economic development, thus defeating some 
of the benefits to be derived from the equitable use of water resources 
within a given locality or region (Jacobs, 1965). 

The Important point which needs to be made here is that the 
concept of stream standards and effluent standards is not mutually ex- 
clusive. In most cases both are necessary. For example, the Illinois 
Sanitary Water Board (1966) has adopted rules and regulations pertaining 
to sewage and industrial waste treatment requirements, effluent criteria, 
and water quality criteria for lakes and rivers, in order to protect the 
water resources of the state. It is true that the amount of weight 



placed on a particular type of standard depends on a given situation, 
the type of waste and the stage of development of the pollution abate- 
ment program. As pollution abatement progress is made and as the de- 
mands for various water uses grow, more and more emphasis will have to 
be given to the stream standards. 

Though the stream quality standards specify minimum accept- 
able levels for a variety of stream quality parameters like bacteria, 
dissolved solids, chemical constituents, etc., the parameter most com- 
monly used as a measure of the pollution from biodegradable waste, by 
investigators concerned with stream sanitation aspects, is the stream's 
dissolved oxygen (DO) concentration. In this study, the primary atten- 
tion will be directed to this parameter, though it is conceptually 
feasible to extend the ideas to deal with other parameters as well. 

With the enactment of the Water Quality Act of 1965, state 

regulatory agencies either have adopted or are in the process of adopting 

(as of May 1968) standards for interstate waters. Among other parameters 

considered for stream standards, the Illinois Sanitary Water Board 

(1966) has adopted different standards for dissolved oxygen in Illinois 

rivers, depending on the water use. Thus, the criteria adopted for 

the aquatic life sector of streams is: 

For maintenance of well balanced fish habitats the 
dissolved oxygen content shall be not less than 5.0 
mg/1 during at least 16 hours of any 24 hour period, 
nor less than 3.0 mg/1 at any time, 

and for the industrial water use sector, It is: 

Not less than 3.0 mg/1 during at least 16 hours of 
any 24 hour period, nor less than 2.0 mg/1 at any 
time. 



Though the standards adopted for dissolved oxygen are defini- 
tive in nature, there is a finite probability that the dissolved oxygen 
is likely to fall below the set standard during 16 hours of any 24-hour 
period and the standard set for the minimum dissolved oxygen at any 
time. There is a growing realisation among the investigators for the 
need of probabilistic stream standards (loucks, 1967; Ledbetter and 
Gloyna, 1964; Thayer, 1966). Loucks (1967) proposes a probabilistic 
stream standard for dissolved oxygen as follows: 

The dissolved oxygen concentration in the stream during 
any 7 consecutive day period must be such that 

1. The probability of its being less than 4 mg/1 
for any 1 day is less than 0,2; and 

2. The probability of its being less than 2 mg/1 
for any 1 day is less than 0,1 and for any 2 or more 
days is less than 0.05. 

OBJECTIVE OF THE STUDY 

In order to design any type of treatment plant, it is necessary 
to know what volume and concentration of pollutant may be discharged to 
the stream so that the stream standards are not violated. This implies 
that there should be methods to predict the response of the receiving 
stream to waste loads placed in it. Several mathematical formulations 
are available for predicting the dissolved oxygen responses, most of 
which are based on the pioneering work of Streeter and Phelps (1925). 
In all these formulations, the reaction velocity coefficients affecting 
the rates of BOD removal, atmospheric reaeration, etc., are taken as 
constants, though it was recognised by a few investigators (LeBosquet 
and Tsivoglou, 1950; Eckenfelder and O'Connor, 1961) that they are far 
from being constants. The main aim of this work is to ascertain the 
significance of the variations in the velocity coefficients K} and K 2 



in defining the dissolved oxygen (DO) response of a receiving stream 
and to develop a procedure for determining the DO taking the variability 
in these rate coefficients into consideration, if these variations are 
significant. 

The specific objectives of this study are: 

1. To examine the relative importance of the variations in 
the reaction velocity coefficients affecting the bacterial oxidation 
of the organic matter and the atmospheric reaeration of river water, 
in predicting the dissolved oxygen responses of the receiving stream. 

2. To determine the nature of variations of these velocity 
coefficients and to formulate and test the hypotheses concerning their 
chance variations. 

3. To develop a procedure for predicting the dissolved 
oxygen in a river downstream of a waste source by taking into account 
the variations in these velocity coefficients. 

4. To enumerate quantitatively the chance variations in dis- 
solved oxygen responses in streams in terms of probability measure. 

The last of the four objectives mentioned above is extremely 
significant in the light of the guidelines established for water quality 
standards under the Water Quality Act of 1965, wherein it is stated that 
numerical values for quality characteristics and the measure of limiting 
values which will govern for purposes of the criteria should be defined. 
The increasing importance of this concept will be felt as and when the 
stream quality standards are implemented and enforced, particularly with 
the advent of growing use of automatic stream monitoring installations. 



SCOPE OF THE STUDY 

The present study is limited to the following: 

1. The relative importance of the variations in reaction 
velocity coefficients on the dissolved oxygen response in the stream 
are evaluated using a hypothetical stream situation based on Streeter- 
Phelps* formulation. 

2. The Ohio River-Cincinnati Pool survey data for the river 
reach between miles 474.6 and 479.05, published by the U. S. Department 
of Health, Education and Welfare (1960) are used for determining the 
values of the velocity coefficients affecting deoxygenation rates in 
river samples. 

3. The data collected by the Tennessee Valley Authority 
(1962) for the prediction of stream reaeration rates are used for 
formulating the hypothesis concerning the chance variation of the 
velocity coefficient governing the rate of atmospheric reaeration 
in rivers. 

4. Statistical models describing the chance variations of 
the two velocity coefficients mentioned above are formulated and tested. 

5. Using Monte Carlo techniques, two sets of data for the 
velocity coefficients are generated on the basis of the statistical 
models formulated. The generated data are tested and verified. 

6. The generated data are used in the conceptual model for 
the dissolved oxygen-biological oxygen demand (DO-BOD) relationship in 
streams for predicting the DO response. The conceptual model is verified 
using the Ohio River-Cincinnati Pool survey data and the probabilistic 
variation of the DO responses are studied. 



II. LITERATURE REVIEW 

The philosophy of protecting the stream for most uses, even 
future uses, is being increasingly accepted, since it is good from the 
standpoint of the water user and the public* It permits planned growth 
of water-using industries and recreation side by side, and it prevents 
undue time lags between the need for water use and its realization. In 
recent years, there has been a great deal of attention towards the 
systems analysis approach to water quality management considering the 
river flow together with the possibilities of affecting or controlling 
the quality of the water at various use points by dams, water purifi- 
cation plants, and waste water and industrial waste treatment plants 
(Thomas and Burden, 1963; Liebman, 1965; Loucks, 1965; Montgomery, 
1964; Worley, 1963). In most of these studies (Loucks, 1965; Liebman, 
1965; Montgomery, 1964; Worley, 1963), the critical dissolved oxygen 
resulting from the addition of organic waste to a water course is taken 
as a controlling parameter for studying the system performance, where 
critical dissolved oxygen concentration is defined as the minimum con- 
centration of dissolved oxygen in the river below a waste outfall. If 
such an approach is to yield more reliable information, it is imperative 
that the DO response of the receiving stream for the waste loads placed 
in it must be predicted with a greater degree of accuracy than is pos- 
sible with the methods adopted in these studies. 

The wastes discharged into a stream from municipal and in- 
dustrial treatment plants contain a large variety of chemical compounds. 
Of primary interest in this work is that portion of the waste which is 



8 

biodegradable and henoe oxygen consuming* When this material is 
introduced in a water course, it undergoes biochemical oxidation, 
caused by microorganisms which utilise the organic matter for energy 
and growth. 

If sufficient dissolved oxygen is present in the water, the 
microflora are primarily aerobic, uti listing the dissolved oxygen to 
carry out oxidation reactions producing water and carbon dioxide as 
end products. If, however, sufficient oxygen is not present, anaerobic 
organisms predominate resulting in undesirable end products. Also the 
concentration levels of oxygen have profound effects on the physiology 
and type of fishes found in the water bodies. The lethal effect of 
low concentrations of DO appears to be increased by the presence of 
toxic substances such as excessive dissolved carbon dioxide, ammonia, 
cyanides, zinc, lead or copper. 

In situations where maintenance of well balanced fish habitat 
is one of the primary objectives, the stipulated standards for dis- 
solved oxygen are that it shall not be less than 5.0 mg/1 during at 
least 16 hours of any 24-hour period, nor less than 3,0 mg/1 at any 
time (Illinois Sanitary Water Board, 1966; Aquatic Life Advisory Com- 
mittee of ORSANCO, 1960), Where other uses like the industrial water 
are more important, the stream standards adopted for such uses with 
respect to DO differ from the above. 

Because of the variety of oxygen -demanding organ! cs in the 
wastes, it is common to measure the strength of wastes in terms of 
their biochemical oxygen demand (BOD) rather than to analyze for the 
chemical constituents of the wastes. 



9 

The most widely used mathematical models for predicting the 
DO responses in a stream are either the one proposed by Streeter and 
Phelps (1925) or the modified predictor equations based on Streeter- 
Phelps' formulation (Camp, 1963; Dobbins, 1964), though a few other con- 
cepts in DO predictions (Churchill and Buckingham, 1956; Thayer and 
Krutchkoff, 1965) have been put forth and applied with varying degrees 
of success* 

The Streeter -Phelps equation considers only two mechanisms 
affecting DO, namely the removal of oxygen by bacterial oxidation of 
organic matter and the absorption of oxygen from the atmosphere. First 
order kinetics are employed to express deoxygenation and reoxygenation. 
The rate of removal of BOD on the basis of first order kinetics is 
given by: 



i - - h 1 (l) 



where L is the ultimate first stage BOD (milligram per liter) remaining 
to be satisfied at any time t (days) and K^ (base e) is the reaction 
rate coefficient, which depends on the characteristics of the waste, 
temperature, type of microorganisms present, and other environmental 
factors. The integration of this equation yields 

-K lt 
L - L a e (2) 

where L a is the initial (t « 0) ultimate first stage demand* Since BOD 

is measured in terms of oxygen consumed, the rate of oxygen consumption 

equals the rate at which BOD is satisfied, i.e., $pL « - ^ - K*L, where 

dt dt x 



10 



D is the dissolved oxygen deficit defined as the difference between 
saturation concentration, C g , at the river water temperature and the 
oxygen concentration, C, in the river. 

Reoxygenation process in the stream due to atmospheric re- 
aeration is also considered as a first order reaction which depends on 
the dissolved oxygen deficit, D, This can be expressed mathematically 
as: 



£ - K 2 (C, - C) - K 2 D - - § ( 3 ) 



where K2 (base e) is the reaction rate coefficient defining the re- 
aeration process* 

Combining the rates of these two reactions, and writing the 
resulting equation in terms of dissolved oxygen deficit, the response 
of the receiving stream for a single waste source is defined by the 
differential equation as: 



f - K lL . K 2 D (4) 



The solution of this differential equation with appropriate initial 
conditions yields the well known and classical Streeter-Phelps equation 
for stream self purification capacity and is given as: 

K l L a 
D " k 2 - K x 0*P <- K l fc ) - «P (-K2O] + D a e *P <- K 2t) (5) 

where D a is the initial dissolved oxygen deficit in the river reach at 
time t « and all other terms are as defined previously. 



11 



Figure 1 shows the typical oxygen sag curve. The point of 
maximum deficit (or minimum dissolved oxygen concentration) is known 
as the critical point* The critical time, or time at which the maximum 
deficit occurs (t c ), and the corresponding maximum or critical deficit 
(D c ) are defined by the following equations as: 

* K2 (K2 - ^i' 
<=c - K7TI7 {m Ck^ (1 - KlLa * D.)]} (6) 

and 

K l 
D c " kJ L a ex P <"*!*<:> < 7 > 

In the river reach prior to the critical point, the rate of deoxygena- 
tion is greater than the reaeration rate and beyond this point, the 
converse holds. The river is said to recover in the latter region from 
the pollutional load discharged into it* 

The shortcomings and criticisms expressed against the Streeter- 
Phelps equation, Eq* 5, for stream self purification capacities and 
attempts of other investigators to modify these concepts in order to 
improve upon the predictions are discussed in the following paragraphs. 
There are also altogether different approaches taken for the prediction 
of DO response of the streams to waste loads placed in it and these are 
discussed along with the suggested modifications for the Streeter-Phelps 
concepts c 

The Streeter-Phelps sag equation when fitted to stream DO and 
BOD data have yielded such a wide range of values of the parameters Kj 
and K« as to suggest that some important factors have not been taken 



12 



•—I c -— 

^^ 

B 

mm* 

u 
m 
u 
u 

s 






o 

(8 
CI 

25 



T 

D a 

* 


; 


, 


D ^"^ 


— 




D 


c 


*'yr 




« tc 

* 


* 


— t 


H 


_______ 



Tine, days 



FIGURE 1. DISSOLVED OXYGEN SAG CURVE 



13 

into account in the formulation or that the variables have not been 
incorporated in the correct mathematical form. Several authors (Camp, 
1963; Dobbins, 1964; O'Connor, 1967) have proposed modifications to 
the Streeter-P helps formulation by taking into account other parameters 
which affect the oxygen uptake in the stream like removal of BOD due 
to adsorption and sedimentation, addition of BOD from benthal layers 
to the overlying water, benthal oxygen demand, oxygen addition due to 
photosynthesis, algal respiration, etc. 

Some apprehension also exists concerning the validity of the 
application of the first order concept and how the results of the BOD 
test should be interpreted mathematically (Gates, 1966; Thomas, 1961; 
Young, 1965). Thomas (1961) assumes second order (bimolecular) reaction 
kinetics for BOD removal, in his "Step Method" for estimating the oxygen 
uptake in a stream. The advantage claimed by the author is that the 
variation of the BOD reaction velocity parameter is considerably re- 
duced if a second order rate equation is used in piece of the first 
order equation. Young (1965) concludes, based on the analysis of the 
data for sewage, that the first and second order kinetics fit the first 
stage BOD data with equal precision at 20°C and 35°C. Since the validity 
of the second order kinetics concept over the first order kinetics has 
not been well established, most of the recent works (Liebman, 1965; 
Loucks, 1965; O'Connor, 1967; Thayer, 1966) use first order kinetics 
for the oxygen depletion due to bacterial degradation of organic 
matter. Inasmuch as this concept is more widely recognized and used 
in the sanitary engineering field, the BOD progression will be assumed 
to follow first order kinetics. 



14 



The merit of employing a mathematical model developed from 

data obtained in such a nondynamic system as BOD bottle and applying 

it directly to a dynamic system such as a stream has been questioned 

in the recent past (Gates, 1966; Gannon, 1963; Nejedly, 1966), Gannon 

(1963) states 

•••it is apparent that no adequate explanation is 
available to account for the differences between 
the laboratory kj and the river k r . This, no 
doubt, is partially due to inadequacies of the 
existing BOD techniques. Probably it may never 
be possible to develop a BOD procedure in the 
laboratory that will duplicate the natural river 
conditions, but certainly efforts should be 
directed toward improving existing procedures. 

In addition to improvement of the laboratory 
BOD test it appears that much useful fundamental 
information could be obtained about BOD removal 
rates from controlled studies using artificial 
channels. This approach, then, would be somewhere 
between the natural stream conditions and the 
highly artificial bottle methods, 

Isaacs and Gaudy (1967) from their studies specifically 
designed to gain an insight into the equivalence of K^ determined 
in the bottle and in a laboratory stream model, using synthetic 
sewage consisting of equal parts of glucose and glutamic acid came 
to the conclusion that data obtained in BOD bottles could be used to 
make a reasonable prediction of the critical DO provided the seed 
concentration in the BOD bottle was identical, or nearly identical, 
to the biological solids concentration in the receiving stream. They 
further observe, based on the model stream study, that a fairly ac- 
curate prediction could be made by estimating the extent of the first 
stage exertion* Since these conditions could be easily met by making 



15 

observations on river samples obtained downstream of a waste source, 
the idea of extrapolating the BOD bottle data to river situations 
cannot be ruled out, at least until a better method could be devised 
and established* 

Nejedly (1966) holds the view that longitudinal mixing, i.e., 
mixing in the direction of flow, induces contacts between particles of 
decaying organic substances with different detention periods within the 
river stretch considered, and hence a more rapid BOD reduction takes 
place in a stream than in a bottle. This view is opposed to those of 
Camp (1965), Dobbins (1964), , Connor (1967) and several others who 
hold that the effect of longitudinal dispersion is negligible in deter- 
mining streams* assimilative capacity* Though, Nejedly presents data 
to support his hypothesis, no quantitative conclusion for the effect of 
longitudinal mixing on the reaction coefficient K could be determined. 

It is generally recognized that the differences in removal 
rates of BOD in BOD bottles and in actual streams are due to the phen- 
omena of biological accumulation and removal of BOD due to sedimentation 
without the DO being affected (Gannon, 1963; O'Connor, 1967). These two 
removal rates are taken as the same in the absence of settleable or- 
ganic solids and fixed aquatic vegetations in the receiving stream 
(O'Connor, 1967). 

In spite of the fact that the bottle BOD technique has been 
subjected to severe criticism, it has been used in one form or the 
other in determining stream assimilative capacities. Recognizing that 
deoxygenation due to bacterial degradation of organic matter is only 



16 



one of the several mechanisms which affect the BOD removal in a 
stream, there seems to be ample justification in asing the data 
obtained in BOD bottles to determine river self purification capac- 
ities, particularly when the mechanisms affecting the BOD removal 
without oxygen consumption are considered separately in mathematical 
modeling* 

Another severe criticism voiced against the use of Streeter- 
Phelps equation or its modifications is that the so-called reaction 
velocity constants Kj, K£, etc, are found to vary considerably. 
LeBosquet and Tsivoglou (1950) observe: 

The velocity constant for reoxygenation, K£, is subject 
to wide variation in different streams, in individual 
stretches of any one stream and for various river 
stages in the same stretch. Although variations in 
K^ are usually of somewhat lesser magnitude, they 
too may be appreciable, and in many situations it 
is quite difficult, if not impossible, to derive 
acceptable values for these velocity coefficients. 

Churchill and Buckingham (1956) state regarding the variability of 

these coefficients as follows: 

The so called constants K^ and K2 are determined 
from an intensive survey, usually at a fairly low, 
steady, flow. However, if another survey is made on 
the same reach, even fairly soon after the first 
one, it at once becomes obvious that the "constants** 
may vary considerably from the values first deter- 
mined .... Therefore, when the derived "constants" 
are far from constant, the results of computation in- 
volving these "constants" for other conditions of 
load, flow, and temperature, are certainly open to 
question. 

The main aim of this work as stated earlier is to ascertain the sig- 
nificance of the variations in the velocity coefficients K^ and K~ in 
defining the DO response of a receiving stream and to develop a procedure 



17 

for determining the DO taking the variability in these rate coefficients 
into consideration, if these variations are significant. 

Velz (1939) presented a method for oxygen balance studies in 
polluted rivers in which the atmospheric river reaeration is computed 
from an empirical relationship, established in a quiescent column of 
water, defining the relationship of the terms: percent saturation 
after a known elapsed time, linear depth of column, molecular diffusion 
coefficient, and the initial percent saturation of the column of water. 
The application of an empirical relationship established in a quiescent 
body of water to a dynamic system as in a river situation is highly 
questionable, particularly when the effect of molecular diffusion of 
oxygen is considered negligible in a stream. Also a family of curves, 
giving the relationship between BOD and time at temperatures 0°C to 
30°C are presented for the purpose of determining the BOD reduction 
along the course of the river. Though first order reaction kinetics is 
assumed in defining these curves, no distinction is made for the differ- 
ences in the characteristics of the wastes under examination, i.e., the 
percent reduction of the pollutional load in a given flow-through time, 
is taken as independent of the nature of the waste. Velz's method 
suffers from the same shortcomings discussed earlier for the Streeter- 
Phelps formulation and in addition this method does not take cognizance 
of the nature of wastes which is an important factor in biodegradability. 

Thomas (1948) discussed a technique for stream analysis which 
recognizes that the K, value may not be 0.23, the commonly accepted 
value in sanitary engineering practice, and also that there may be BOD 



18 



sinks other than. that represented by bio-oxidation. Furthermore, his 
discussion indicates that he did not consider these additional BOO 
sinks to be oxygen sinks. This technique requires that, through the 
use of the BOD test, the values of K^ and ultimate first stage demands 
be determined at boundary stations of a hydraulically uniform stretch 
of stream. The difference between the ultimate first stage BOD values 
of the upstream and downstream stations represents the reduction in 
oxygen demand realized through the reach. This information combined 
with the time of passage, and assuming first order kinetics, is used 
to determine the rate constant expressing the reduction in potential 
oxygen demand by the biomass is determined by averaging the K^ values 
observed for samples from the two boundary stations. The difference 
between this rate and the rate observed for the stream represents the 
rate at which potential oxygen demanding material is going to non- 
oxygen consuming sinks. 

Although this approach recognizes that Kj may vary, it assumes 
that the stream values of K. and L can be predicted from bottle measure- 
ments. The concept that not all potential oxygen consumption need 
necessarily be satisfied by actual consumption was a significant con- 
tribution to stream analysis. But for this modification, Thomas' tech- 
nique is identical to that employed by Streeter and Phelps (1925) and 
subject to the same previously-indicated reservations. 

LeBosquet and Tsivoglou (1950) published data for the Ohio 
River indicating a radically different method of pollutional analysis 
employing statistical methods. They established a linear relationship 
between critical DO deficit in mg/1 and reciprocal of river flow in 



19 

cubic feet per second, with a high degree of correlation. This method 
permits direct analysis of oxygen-flow relationship in a given stretch 
avoiding the use of factors such as Kj and K 2 . The method outlined by 
them cannot give better results than the oxygen sag formula since the 
characteristics of the wastes and the reoxygenation capacities of the 
stream fluctuate greatly and such variations will introduce inaccuracies. 
It is based mainly on the surmise that the effect of the variations in 
K^ and K 2 are relatively minor, which is not a valid conjecture as will 
be borne out by this study. 

Churchill and Buckingham (1956) extended the concept of 
LeBosquet and Tsivoglou (1950) and presented an analysis employing 
statistical techniques of multiple correlation and giving the maximum 
deficit in terms of a linear equation with 5-day BOD, water temperature 
and stream discharge as variables. The advantages claimed for this 
method are that evaluation of factors like K^, K 2 , L and flow-through 
time are avoided and that the correlation analysis predicts the critical 
value reasonably well. 

The authors (Churchill and Buckingham, 1956) recommend 
choosing a station for making 5-day BOD observations downstream of 
the pollution, and as close to the critical section as possible. Since 
the critical section in the river fluctuates with variations in stream 
flow, and the characteristics of the waste, the criteria suggested by 
the authors for selection of site for making BOD observations to be 
used in the regression analysis is not easily met since it is not 
feasible to estimate the location of the critical section from their 
statistical model. Also, it is not possible to predict the state of 



20 

oxygen concentration in different sections of the river using the 
correlation technique* It is of paramount importance to estimate 
assimilative capacity at any desired reach and not merely the critical 
reach of the river, particularly when comprehensive river basins 
planning are envisaged with planned development of rivers. 

Thomas (1961) presented a method called the "Step Method" for 
estimating oxygen uptake in a stream. Like the St reeter-P helps formu- 
lation, only two mechanisms affecting DO in a stream are considered, 
namely the removal of oxygen due to bacterial decomposition and addition 
of oxygen from the atmosphere, even though he has emphasised earlier 
(1948) the importance of considering removal and addition of BOD due to 
sedimentation and scour respectively. The only departures from the 
classical Streeter-Phelps formulation are that the bacterial degradation 
of organic matter is assumed to be governed by second order kinetics 
and that the values of 5-day BOD are used instead of the ultimate first 
stage BOD. In the light of the investigations of Young and Clark 
(1965). and Isaacs and Gaudy (1967) discussed earlier, the "Step Method" 
does not appear to improve the predictive techniques for stream self 
purification capacities. 

Camp (1963). and Dobbins (1964) proposed equations which are 
very similar, for predicting dissolved oxygen deficit by taking into 
account benthal demand, removal or addition of BOD due to deposition or 
scour, and photosynthesis. , Connor (1967) further extended these ideas 
by considering the diurnal variation of oxygen addition due to photosyn- 
thesis instead of treating it as time constant for the whole 24-hour 
period as in Camp's model, and adding another term for algal respiration 



21 

in the oxygen sinks. Since these formulations (Camp, 1963; Dobbins, 
1964; O'Connor, 1967) are only modifications of the basic Streeter- 
Phelps formulation, considering a few more plausible mechanisms 
affecting the DO-BOD relationship in streams, they do not obviate 
all the criticisms put forth for the Street«r-P helps equation. 

Thomann (1963) postulated a mathematical model using the 
systems analysis concept for predicting the response of any water 
quality parameter, affected either by conservative or nonconservative 
pollutants. The model is developed for a one-dimensional estuary for 
which stream analysis is only a particular case. The model assumes 
that a body of water can be segmented into a discrete number of parts 
each of which is treated as a stationary linear subsystem forming part 
of the whole system. The time rate of change of a water quality parameter 
particularly DO, in any of the subsystems is taken as affected by the 
advection and diffusion of DO both into and out of the system and by 
the sources and sinks of DO within each subsystem. These additions or 
removals of DO are considered as the variables that force the DO to 
respond. An equation representing statinary linear subsystem for DO 
concentration at a given time, with the sources and sinks of DO as 
forcing functions and also with feed-back mechanisms from adjoining 
subsystems, is developed. Since the subsystems that form the overall 
system are all linear, the solutions obtained for each subsystem are 
linearly superimposed to predict the response of the system as a whole. 

Though Thomann (1963) has indicated conceptually a method for 
predicting the DO response when the decay coefficient for organic matter 
changes with time, no mention is made in his theoretical development, 



22 

how the variations with tiiae in the decay coefficients are evaluated* 
Moreover, the possible variations in the reaeration coefficient are 
overlooked. It is well documented (TVA, 1962) that the maximum re- 
aeration coefficient observed under constant flow conditions in the 
Tennessee Valley rivers, when the temperatures remained nearly uniform, 
ranged approximately from 2 to 9 times the minimum values depending on 
the characteristics of the river, Thomann's work marks a significant 
advance in pollution analysis wherein an attempt has been made to con- 
sider all the factors affecting DO in a body of water taking into account 
the possible variations in some of the parameters affecting the system. 

Thayer (1966) viewed the oxygen relationship in streams as a 
stochastic birth and death process with the BOD and DO being increased 
and decreased by small amounts in a very short interval of time. The 
stochastic model provides for the joint density function for both pol- 
lution and dissolved oxygen for different initial conditions. The ad- 
vantage claimed by the author is that in addition to predicting the mean 
DO concentration which is the same as that predicted by the determin- 
istic equation of Dobbins (1964), it affords a measure of the variance 
of DO from its mean value, thus enabling one to predict the probability 
of DO downstream of a source of pollution taking any given value. 

In developing his model, Thayer (1966) did not consider the 
effects of photosynthesis and algal respiration. The prediction equation 
was verified with the data published by the Resources Agency of California 
(1962) for the Sacramento River. Though the published data indicate a 
value for K. ranging from 0.12 to 0.55 and emphasise the importance of 



23 



photosynthesis and algal respiration in the river reach below Sacra- 
mento, these effects were totally ignored in the model verification* 
Also in the controlled laboratory experiments with dextrose as the 
only substrate for simulating river conditions, values ranging from 
0.076 to 0.167 were obtained forKj (Thayer, 1966). Whereas, in the 
verification of his theory with the results of the laboratory data, a 
value of 0.07 for Kj has been assumed on the supposition that the as- 
sumed value is reasonably close to the observed values. In the opinion 
of this writer, the verification of the mathematical model (Thayer, 
1966) either with the published data for the Sacramento River or with 
the laboratory data leave much to be desired. 

From the foregoing general discussion on the present state 
of knowledge concerning the stream assimilative capacities, it can be 
summarised that though there are strong indications from field measure- 
ments for the variability in the parameters affecting DO-BOD relation- 
ship, and though a need for talcing into account these variations is 
strongly felt, all the theories so far postulated except that of Thomann 
(1963) fail to consider this aspect. Even though there is considerable 
criticism for extrapolating the BOD bottle observations to river con- 
ditions, this is the only expedient way available at the present time 
for defining the biological interactions in the environment. The con- 
cept of first order kinetics, first introduced by Streeter and Phelps 
(1925), with the parameters determined by bottle BOD tests will continue 
to be used until a better and more reliable method is evolved to sub- 
stitute this concept. 



24 



III. SENSITIVITY ANALYSIS 

DEFINITION 

Sensitivity analysis applies to the concept in which all 
except one of the variables in a system are held constant and this 
single exception is allowed to vary through its full possible range. 
The effects on certain system performance are noted, testing the 
sensitivity of the system to that particular variable which is varied. 
If the measured output varies significantly, the system is said to 
be sensitive to that variable for the given conditions. In a mathe- 
matical sense, the slope of the output as a function of input is a 
sensitivity measure. 

Two variables of a system may be permitted to vary simultan- 
eously, each having an effect on the output measure. Mathematically 
this becomes a three-dimensional model; the output function becomes a 
surface rather than a line. This is called a response surface. A well 
defined response surface contains outcomes of all possible combinations 
of the two variables. In a mathematical sense, the sensitivity of 
the response surface to a single independent variable is a partial 
derivative of the response surface equation with respect to that variable. 

PROCEDURE 

The suggested general procedure for carrying out a sensitivity 
analysis is as follows: 

1. The independent variables of the system are listed and 
the possible range of values for each of the variables is ascertained. 



25 

2. The value of the response surface variable is calculated 
using nominal values. This value serves as a reference point* 

3. Percent change in the response surface is computed by 
substituting the low value of one of the independent variables, 

4. The low value is replaced with an incremented value for 
the same variable as in step 3 and the percent change in the response 
surface is evaluated. This is repeated until all the values within 
the possible range of values for the variable under consideration are 
covered. 

5. Steps 3 and 4 are repeated for each of the independent 
variables which is likely to assume a range of values instead of a 
single value. 

The above procedure for individual variables analysis can be modified 
to test for changes of variables in groups which may be related. 

Nominal Values in the Streeter -Phelps Equation 

The Streeter -Phelps equation for stream assimilative capacity 
(Eq. 5) will be used in the sensttivity analysis for determining the 
significance of the variations in reaction velocity coefficients K. and 
Kg in predicting the dissolved oxygen response. It is generally ac- 
cepted in sanitary engineering practice that the average value for K^ 
at 20°C for domestic wastes is 0.23 per day (base e) (Fair and Geyer, 
1954; Sawyer, 1960), even though values ranging from 0.1 to 0,7 per day 
are indicated by some authors for domestic and industrial wastes (Camp, 
1963; Eckenf elder and O'Connor, 1961). The Kj^ value at 20°C reported 
for river samples collected downstream of Sacramento sewage treatment 



26 

plant discharge is found to range from 0,12 to 0.55 per day with an 
average value of 0.294 per day (The Resources Agency of California, 
1962). The computed Kj values for the Ohio River samples obtained 
during a 1957 survey (USPHS, 1960) in the river reach between miles 
474.6 and 479,05, where all the pollution entered prior to the reach 
under consideration, showed a range from 0.05 to 0.32 per day with an 
average value of 0.173 per day. From the average value cited for 
K. in the literature and from the values of K obtained in actual 
river surveys, it can be concluded that the nominal (average) value 
for the purpose of sensitivity analysis can be taken as 0.23 per day 
at 20°C. 

The nominal value for K 2 at 20°C is not very well defined 
since it depends on the characteristics of the stream* The classifi- 
cation of streams from reaeration point of view is only subjective in 
nature. Fair and Geyer (1954) indicate for large streams a value of K2 
(base e) at 20°C ranging from 0,4 to 0.7 per day. Eckenfelder and 
O'Connor indicate a common range of K 2 from 0.2 to 10.0 per day, the 
lower value representing deep slow-moving rivers and the higher value 
for rapid shallow streams. In one set of experiments (No. 14) in the 
Holston River, which could be classified as a large river with moderate 
velocity, K 2 had a range of values from 0,10 to 1.18 per day with an 
arithmetic average of 0.63 per day under comparable conditions within 
practical limits (Buckingham, 1966), Similar ranges of values were ob- 
tained in a few other experiments conducted in TVA rivers. Hence an 
average value, in the case of large streams, for K 2 equal to 0.6 per 
day at 20°C will be a realistic assumption. 



27 



Temperature Effects on Reaction Coefficients 

The Importance of temperature effects on the velocity coef- 
ficients K, and K2 has long been veil recognized and much attention 
has been focused on this aspect. Streeter and Phelps (1925) described 
the relationship between K^ and temperature by the expression 

K 1(T) " K 1<20)9 T " 20 (8> 

in which Kj^ T ) is the deoxygenation coefficient at any temperature T 
in degrees centigrade, Ki(20) * s tne deoxygenation coefficient at 20°C 
and 6 is the temperature coefficient* 9 was found to have an average 
value of 1.047, Theriault, as reported by Camp (1963), and Thomas (1961), 
confirmed this value of 9, based on his Ohio River studies. Recently, 
Zanoni (1967) in an attempt to establish the temperature effects on 
the rate of deoxygenation of a conventional activated sludge waste 
water treatment plant effluent, found the value of 9 to be 1.048 for the 
temperature range of 10 to 30°C. Though the observed values for 9 
varied within a small range, the effects of this variability in estimating 
the deoxygenation rate at other temperatures, knowing its value at 
20°C, will not be significant. This temperature relationship with 9 
having a value of 1.047 will be used in computing the nominal values 
at other temperatures, as has been employed in earlier works (Ecken- 
f elder and O'Connor, 1961; 0*Connor, 1960; Streeter and Phelps, 1925; 
Thomas, 1961; Worley, 1963). 

Also, it has long been known from experimental evidence that 
deoxygenated water will absorb oxygen from the atmosphere at a higher 



28 

rate if the temperature of the water is raised, other conditions being 
held constant. In the normal range of stream temperatures, a rise In 
water temperature results in a decrease in viscosity, density, and 
surface tension. It is difficult to distinguish the exact role played 
by each of these factors. Because it is only the net overall effect 
of temperature on re aeration that is of interest in most investigations, 
experiments have been conducted in the past for measuring reaeration 
rates at several temperature levels. Results of such experiments have 
been generally reported in the following form: 

K 2(T) - K 2(20)* T " 20 <*) 

in which K2( T ) is the reaeration coefficient at temperature T in degrees 
centigrade, K2(20) * s the reaeration coefficient at 20°C and $ is the 
temperature coefficient. Several authors indicated values for $ ranging 
from 1.008 to 1.02 as reported by the Committee on Sanitary Engineering 
Research in their Thirty-first Progress Report (1961). The committee 
(1961) concluded, after conducting a series of carefully controlled 
experiments that the thermal coefficient $ remains constant over a 
wide range of turbulence conditions, with a value of 1.0241. The re- 
lationship shown in Eq. 9 with $ equal to 1.0241 will be used in com- 
puting the nominal values at other temperatures. 

Table 1 shows the nominal values of the reaction velocity 
coefficients Kj and K 2 at temperatures 10°C, 20°C, and 30°C used in 
the sensitivity analysis. 



29 



TABLE 1 
NOMINAL VALUES OF REACTION COEFFICIENTS K t AND K 2 



Parameters 




Nominal Values 






At 10°C 


At 20°C 


At 30°C 


K l 
K 2 


0.15 
0.47 


0.23 
0.60 


0.36 
0.76 



The range of values considered for Kj^ and K 2 are 0.1 to 0.6 
per day and 0.2 to 3,0 per day respectively with incremented values 
of 0.1 per day for Kj and 0.2 per day for K 2 . The initial conditions 
assumed for ultimate BOD and DO deficit in the hypothetical stream at 
the source of pollution are 20 mg/1 and 1.5 mg/1 respectively, which 
are considered as realistic values for these parameters, having been 
based on actual observations made on river samples (Gannon, 1963; 
USPH5, 1960). Since the effect of temperature on ultimate BOD is 
considered insignificant (Gotaas, 1948; Thomas, 1961; Zanoni, 1967), 
no correction for temperature effects will be made for estimating 
initial ultimate BOD at different temperatures considered in the 
sensitivity analysis. 

RESULTS 

The critical DO predicted by the Streeter-Phelps equation is 
considered for evaluating the response of the hypothetical stream 
system. The value obtained for critical DO, when K^ and K 2 take their 
respective nominal values, is treated as the nominal value of critical 
DO for the given conditions of initial BOD, DO deficit, and temperature. 
Percent deviations of the system responses are computed when K^ and K 2 



30 

take different values within the possible range and these are based on 
the reference value for the critical DO in the stream, 

Eq, 5, which expresses the relationship between D, L a , D a » t, 
K^ and K 2 , becomes indeterminate when K^ and K 2 are equal. Since the 
ranges of values, which these two coefficients can assume, overlap it 
is necessary to predict the critical deficit when Kj equals K 2 . The 
solution for the Streeter-Phelps formulation, Eq, 4, is given below for 
the case when K^ and K 2 are equal. 

D - K x L a t exp (-l^t) + D a exp (-K L t) (10) 

The time of flow for critical deficit, and critical deficit 
are: 

L a - D a 
'c - "kIET (ID 

D c - L a exp (-K x t c ) (12) 

A computer program incorporating the general Streeter-Phelps equation, 
and the solution for the situation when K^ equals K 2 was written in 
FASTRAN langaage for the IBM 7094-1401 system to evaluate the state of 
oxygen concentration at 1, 2, 3, 4, and 5 days of flow -through time in 
order to trace the oxygen sag profile. The program computes the flow- 
through time for critical deficit and hence the critical deficit and 
critical DO for different combinations of values for K} and K 2 with the 
assumed initial conditions for L a and D a . The flow diagram for the 
sensitivity analysis computer program is shown in Appendix B. Knowing 
the critical DO in the stream, it is then possible to reckon the percent 



31 

deviations in the predicted DO from the reference value when one of 
the reaction coefficients takes different values within its practical 
range of values while the other reaction coefficient assumes its 
nominal value and all other parameters are held at their respective 
constant values* 

Tables 2 to 4 show the sensitivity of the hypothetical river 
system for the variations in K 2 at 10°C, 20°C, and 30°C respectively 
for the assumed initial conditions of BOD (20 mg/1) and DO deficit 
(1*5 mg/1). These are shown plotted in Figure 2. When K 2 takes the 
nominal value at a particular temperature, there will be no error in 
predicting the response of the system* However, when the actual value 
of K2 is lesser or greater than the nominal value, the actual value for 
critical DO will be different from its nominal (reference) value* The 
figure clearly shows the temperature dependence of the sensitivity of 
the system for the variations in K 2 * The error in prediction increases 
with increased deviations of the coefficient K 2 and this becomes in- 
creasingly significant at higher temperatures. Also it should be noted 
that there are upper bounds for these errors when the actual values for 
K 2 are lesser or greater than the nominal value* When the value is far 
less than the nominal value* the rate of deoxygenation exceeds the rate 
of reaeration with the result that the critical deficit could reach its 
limiting value equal to the DO saturation concentration for the given 
temperature* In such a case* maximum deviation could only be 100 per- 
cent, and this effect is seen for the situation when K 2 equals 0*2 per 
day at 30°C. On the other hand when K„ has a value considerably greater 



32 



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0,0 0.5 1.0 1.5 2.0 2.5 

{(.•••ration Coefficient, K 2 , p«r day 



3.0 



FIGURE 2. EFFECT OF VARIATIONS IN K, VALUES ON PERCENT 

ERROR IN PREDICTING THE CRITICAL DO AT DIFFER- 
ENT TEMPERATURES 



36 



than the assumed average value, the rate of reaeration will be con- 
siderable compared to the deoxygenation rate with the result that 
the critical deficit in the system vill be the initial deficit it- 
self* This upper bound for the percent error in DO prediction varies 
with the temperature. 

Table 5 shows the sensitivity of the system for the vari- 
ations in Kj at 10°C, 20°C, and 30°C. These are plotted in Figure 3. 
Figures U to 6 show the relative effects of the deviations in K^ and 
K£ on the sensitivity of the system at temperatures 10°C, 20°C, and 
30°C respectively. The curve marked as "lower scale" indicates the 
sensitivity of the system when K2 varies and all other parameters of 
the system remain constant. Likewise the curve marked as "upper scale" 
refers to the sensitivity of the system when K, varies while other 
parameters remain constant. It is seen from these curves that except 
at 10°C, the variations in the velocity coefficients are equally sig- 
nificant in predicting the critical DO. At 10°C, the error in prediction 
when Kj takes values greater than its nominal value is much more than 
the error due to the variations in K2» 

Generally, it is seen that the errors in prediction due to 
the variations in the reaction coefficients are considerable, with the 
result that there is no justification for using average values for 
these parameters in predicting dissolved oxygen concentrations. Hence 
it is postulated that by treating the velocity coefficients as variable 
coefficients Instead of as constant coefficients, the values predicted 
by mathematical modeling can be made to approach the true value for DO 



37 



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38 



100 



90 



80 



70 



60 



50 



40 



30 



20 



10 




K2 ■ 0.76 per 
day, 30«C 

K- ■ 0,60 per 
day, 20°C 

K, - 0.47 per 
diy, 10°C 







0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 
D* oxygenation Coefficient, K^ f per day 



FIGURE 3, EFFECT OF VARIATIONS IN K & VALUES ON PERCENT ERROR 
IN PREDICTING CRITICAL DO AT DIFFERENT TEMPERATURES 



u 
u 

o 



4J 



I 



u 
hi 

a. 



80 



70 



0.0 



• 

3 

m 

> 

* 60 1— 

« 
c 

8 

z 

8 

u 

8 



50 



40 



30 



20 



10 — 



^oxygenation Coefficient, K 1# per day 
0.1 0.2 0.3 0.4 0.5 



39 



0.6 



Upper Scale 
K 2 - 0.47 per 
day, 10°C 




Lover Scale 
K t - 0.15 per 
day, 10°C 



0.0 0.5 1.0 1.5 2.0 2.5 
Reaeration Coefficient, K 2 , per day 



3.0 



FIGURE 4. COMPARISON OF THE EFFECTS OF VARIATIONS IN K 1 AND 
K 2 ON PERCENT ERROR IN PREDICTING THE CRITICAL DO 
AT 10°C 



40 



Deoxygenation Coefficient, Kj, per day 



90 



0-fl. 



80 



e 

s 

« 
> 

4 
ti 

8 

ss 

s 

u 



u 
u 



o 

§ 
* 

i 

8 



70 — 



60 



50 — 



40 — 



30 



20 — 



10 



Y 



■ fl T L 



0.3 



0.4 



0.5 



Upper Scale 
K 2 - 0.60 per 
day, 20°C 



0.0 0.5 1.0 1.5 2.0 2.5 
Reaeration Coefficient, K^? per day 



.0.6 




3.0 



FIGURE 5. COMPARISON OF THE EFFECTS OF VARIATIONS IN K t AND 
K„ ON PERCENT ERROR IN PREDICTING THE CRITICAL DO 
20°C 



a 



41 



160 



0.0 



140 



• 

3 

m 
> 

* 120 

e 

I 

as 
I 

2 ioo 



8 



► 

u 

S 

8 

« 

a, 



80 



60 



40 



20 



Deoxygenetlon Coefficient, Kj, per day 
0.1 0.2 0.3 0.4 0.5 



pper Scale 
K 2 » 0.76 per 
day, 30°C 



0.0 0.5 1.0 1.5 2.0 2.5 
Reaeratlon Coefficient, K-, per day 



0.6 




3.0 



FIGURE 6. COMPARISON OF THE EFFECTS OF VARIATIONS IN Kj AND 
K, ON PERCENT ERROR IN PREDICTING THE CRITICAL DO 
AT 30®C 



42 



better than it is possible now. The problem of evaluating the 
assimilative capacities of streams is approached by simulation 
technique using the Monte Carlo method which makes possible the 
treatment of the parameters Kj and K2 as variable coefficients. 
The concepts and procedure involved in these will be developed in 
subsequent chapters. 



43 



IV. PROBABILISTIC VARIATIONS IN 
VELOCITY COEFFICIENTS 



It has been found that in many of the practical problems 
which defy theoretical solutions, an approximate but workable solution 
could be found using Monte Carlo techniques. In these methods, it is 
necessary to specify a priori the probability distributions for the 
random variations of separate events which are then merged into a 
composite picture. Few attempts have been made to define the chance 
variations, in terms of probability measure, for the initial BOD and 
DO at the source of pollution. Montgomery (1964) assumed the daily 
variations of waste loads, in terms of BOD, to be normally distributed 
and the assumption was based on considerations given to the histories 
from Gary, Indiana, Galesburg, Illinois, Dallas, Texas, and Racine, 
Wisconsin. Whereas, Thayer (1966) in an attempt to formulate a stochastic 
model for pollution and dissolved oxygen in streams postulated, without 
verifying the assumption, that the initial BOD and DO in a stream vary 
independently according to the binomial distribution law. Though it is 
conceptually feasible, under certain simplified assumptions, to consider 
all the separate events involved in the mathematical model for stream 
assimilative capacities as random events, attention will primarily be 
focused on the variations in velocity coefficients K^ and I<2 and the 
proposed probabilistic model will be tested for known initial conditions. 

VARIATIONS IN THE DEOXYGENATION COEFFICIENT 

As has been mentioned earlier, it is well documented that the 
deoxygenation coefficients of domestic and industrial wastes at any given 
location vary considerably and attempts have been made to formulate 



44 



methods to circumvent the difficulties and inaccuracies arising out 
of these variabilities (Churchill and Buckingham, 1956; LeBosquet and 
Tsivoglou, 1950), In order to place the art of predicting stream 
assimilative capacities on a more rational and time tested basis than 
those proposed by Churchill and Buckingham (1956), and LeBosquet and 
Tsivoglou (1950), it is necessary to extend the concepts of Streeter 
and Phelps (1925) taking variabilities in reaction coefficients into 
account • 

In sanitary engineering practice, it has been tacitly assumed 
that the characteristics of any waste, particularly those of domestic 
origin, vary from time to time. In most of the controlled studies, it 
is customary to use synthetic sewage rather than the waste generated in 
an actual situation. One of the parameters which define the character- 
istics of a waste is the rate at which it is biologically oxidized. 
Since the characteristics of wastes vary considerably from time to 
time, the rate parameter for the wastes, treated or untreated, also 
vary considerably. The general course of deoxygenation and the K^ values 
vary since biological processes deal with the response of living or- 
ganisms to their environment and do not show the same degree of uni- 
formity or consistency even under standard conditions, as purely 
chemical processes do. Even though, this phenomenon can be attributed 
to the viability and type distribution of microbial population present 
and the changes in characteristics of the substrate for the microbes, 
it is not possible to predict the inter-relationship of changes in 
waste characteristics and microbial response quantitatively in a deter- 
ministic sense. Due to the inherent ambiguities involved in the 



45 



deoxygenation process, where most of the contributing factors are 
subject to random fluctuations, it is very well suited for being 
treated under the theories of probability* 

Ki Values of the Ohio River Samples 

Whenever one is inferring general laws on the basis of 
particular observations associated with them, the conclusions are 
uncertain inasmuch as the particular observations are only a more or 
less representative sample from the totality of all possible ob- 
servations. One way of reducing uncertainty in a solution is to collect 
and base it upon more observations. Data from a 1957 survey of the 
Ohio River, collected and published by the United States Public Health 
Service (1960) in connection with the evaluation of the effects of 
navigational improvements on assimilative capacity of the Cincinnati 
Fool, is one of the most extensive and comprehensive set of all river 
pollution survey data published. Extensive information on long-term 
(20°C) BOD test results for sewage treatment plant effluent and river 
samples downstream of effluent outfalls, DO test results, photosyntheti c 
oxygen production ("Light and Dark Bottle") test results, etc., are 
presented. For purposes of studying the probabilistic variations of 
the rate coefficient Kj in an actual situation, the long-term BOD data 
on. river samples collected in the reach between miles 474.6 and 481.45 
is used in this study. All the pollutional loads emanating from Cincin- 
nati enter the river above this reach. The long-term BOD data on river 
samples are abstracted from the published data and presented in 
Appendix D. 



46 



Having the BOD data available, the next step was to compute 
the monomolecular curves which best fit the observed data* It was 
noted earlier that, because of its traditional usage and general ac- 
ceptance, it will be assumed that the first stage of deoxygenation 
proceeds according to a monomolecular or first order reaction* Some 
method had to be selected of the numerous ones reported in the liter- 
ature for the evaluation of the doxygenation coefficient and the corres- 
ponding ultimate BOD* 

One of the earlier methods for the calculation of the velocity 
coefficient K^ was that proposed by Reed and Theriault (1931) which 
uses the least squares principle for minimizing the residual errors 
between the observed data and the fitted monomolecular curve* In the 
application of the procedure, a method of successive approximations 
involving trial and error is used in which it is necessary to assume a 
trial value of Kj_, and this is then used together with the observed 
data in determining a correction factor to the assumed Kj value* The 
operation is repeated with the corrected Kj value until the correction 
factor is small, or in other words, until the assumed and calculated 
values of K. approach one another* Obviously if the initial estimate 
is good, the number of iterations will be minimal, but even if this 
estimate is poor the procedure will eventually yield a good estimate 
of K 1# The time necessary for hand computation is excessive and this 
no doubt had been a factor in the development of less rigorous com- 
putational and graphical procedures (Fair, 1936$ Moore, Thomas, and 
Snow, 1950; Thomas, 1937)* Since it is generally considered (Gannon, 
1963; Zanoni, 1967) that the Reed -Theriault method affords the most 



47 



reliable estimation for K^ and ultimate first stage BOD, this method is 
used in this study in evaluating these parameters for the observed BOD 
data. The flow diagram for the computer program is shown In Figure 7, 
and the tolerance limit between the assumed and corrected values for K| 
is set at 0.005 per day. The terminology used in the flow diagram and 
in the computer program is the same as that adopted by Reed and Theriault 
(1931). 

In determining the parameters for the first stage carbonaceous 
demand, it is necessary to make corrections for the initial lag in 
oxygen uptake and oxygen demand due to nitrification, if these effects 
are significant in the BOD progression data. Figure 8 shows the long- 
term (20°C) BOD data for effluent samples from Mill Creek and Little 
Miami sewage treatment plant. It is readily seen that there is no lag 
in initial oxygen uptake in either of these samples. Also, the trend 
in BOD progression deviates on or around the tenth day, heralding the 
onset of nitrification in the bottle tests. Figure 9 shows the long- 
term (20°C) BOD progression results for the river samples obtained in 
the Ohio River-Cincinnati Pool area. Here again, no noticeable lag is 
discerned and the nitrification process is found to set in on or around 
the tenth day. The BOD progression curves presented here are typical 
of the results obtained in the Ohio River survey. Since the flow-through 
time involved in the survey is never more than four days, oxygen demand 
due to nitrification is not considered in this study. Also, the BOD 
progression data observed up to the tenth day were used in the Reed- 
Theriault method for computing Kj and ultimate first stage BOD values. 
In most cases there were six observations within ten days in each series 



48 



& 



Read input values for BOD (Y), time (t) 
days, estimated value for K^ y and tolerance 
limit for correction factor. 



Write output 
values of L 
and K, 



I - 1 



f^ - 



Compute 
[l - exp (-Kjt)] 
t exp (-Kjt) 



Ef 2 ' 



"i T 



Ef iLi 



r£l£2. 



zf?v 



If 2 ^ - (Ef 1 f 2 )' 



H 



Correction factor, H 
Zf x 2 E f 2 Y - Efjf, 



EfiY. 



Ef. 



Efo Z - (E f 



"7TT-J 





K i^L 



Write error message « 



Yes 




End 



FIGURE 7. FLOW DIAGRAM FOR COMPUTING K, AND ULTIMATE FIRST STAGE BOD 

VALUES USING REED-THERIAULT METHOD 



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51 

of BOD progression tests and five observations in a very few cases. 
The values for Kj (20°C) and ultimate first stage BOD computed for 
each series of observations on the Ohio River samples are shown in 
Appendix D. 

Randomness Test 

Many nonparametric statistical procedures are based on the 
assumption of a random sample of univariate observations. Also 
randomness of a set of observations is sometimes itself the property 
investigated. Although the concept of independent observations, a 
random sample, is well-known, it is worthwhile to define it here. A 
sample of n observations, Xj, %2» •••» x n from a population with dis- 
tribution function F(x) is called a random sample from that population 
if 

P {x 1 ,x 2 ,...,x n } - P { Xl } P {x 2 } ... P {xj (13) 

where P {•} is the probability of observing a value less than or equal 
to the stated value. When a set of observations is analysed, the first 
step is to investigate whether the observations may be regarded as a 
random sample or not. The next step is to formulate a hypothesis for 
the type of distribution of the population from which the sample is 
supposed to be drawn randomly and to estimate values of the parameters 
of the assumed distribution based on the sample. 

The most common case attempted is that showing the relationship 
of an effect to many causes of which a small number of the causes exert 
greater influence than do all others. In such a case, when neglected 



52 



vari Abies, inherent errors, and nonhomogeneity of data have relatively 
small effects, the relationship between the remaining limited number 
of variables would indicate a narrow spread around a basic function, 
Stoltenberg and Sobel (1965) in their study to determine the effects 
of sampling location, organic loading, tidal stage, and sample col- 
lection temperature on K^ values for various samples obtained in the 
Delaware Estuary, came to the conclusion based on analysis of covari- 
ance that these factors did not have significant effect on K^ values. 
Hence it is hypothesized that the variations in K^ values are generally 
random, not attributable to any definite cause with certainty and 
this conjecture was subjected to statistical randomness tests using 
the observed K. values for the Ohio River samples. If the hypothesis 
fails, then one has to separate the trend in the series of observations 
made and study the random variations about the observed trend. 

Determination of Sequence Order . The first step in applying 
a test of the random sample hypothesis consists of placing the n ob- 
servations considered in an appropriate sequence order (Walsh, 1962). 
The method of assigning the integers l,...,n to the n observations 
depends on the experimental situation considered. A common method of 
assignment is on the basis of the times at which the observations are 
produced. Then the first observation produced is assigned the integer 
1, etc. The observations can be ordered on the basis of location. 
Also a combination of location and time can be used. In the Ohio 
River survey, samples from the river reach under consideration were 
collected for analysis from five locations starting from the head end 



53 

of the reach and moving downstream to cover all the locations and the 
whole operation was repeated several times. Since the fate of pol- 
lution and the condition of oxygen concentration in the river as the 
waste discharge flows along the river are of primary importance, the 
observations for K. values are ordered on the basis of the times at 
which the observations are produced. The sequence of observations 
thus obtained was then subjected to randomness tests which are dis- 
cussed in the following sections. 

Another approach would be to group the observations according 
to the place of observation so that any decreasing trend in the Kj 
values could be discerned and isolated. It is likely that the Kj values 
of river samples decrease in the downstream sections as the more 
readily degradable organics are oxidized. This approach is not con- 
sidered in this study since the river sections from which samples were 
obtained in the Ohio liver (USPHS, 1960) are not separated far enough 
to give any meaningful result. 

Runs Up and Down Test . Considering a sequence of n different 
observations x^, X2»...,x n and the sequence of signs (+ or -) of the 
(n - 1) differences (x... - x. ), a sequence of successive + signs is 
called a run up and a sequence of successive - signs a run down (Hald, 
1960; Walsh, 1962). The length of a run is given by the number of same 
signs defining the run. The total number of runs is denoted by R, the 
number of runs of length i by r^ and the number of runs of length k or 
more by R k where 



n - 1 

Z 
i - k 



R k - Z r i (14) 



54 



The expected value for the total number of runs E(R), and the variance 
In the number of runs V(R) are given by 

E(R) - i (2n - 1) (15) 

and 



V(R) - ^ (16n - 29) (16) 



Also for n > 20, R may be regarded as normally distributed 
with good approximation* The expected number of runs up and down of 
length k or more in random arrangements of n different observations for 
values of k from 1 to 7 are given by Hald (1960) in Table 13.10. 

If the hypothesis alternative to the hypothesis of randomness 
is that a gradual change in the level of distribution has taken place 
during the drawing of the samples, such a change will produce trends 
or cycles in the observations, so that one or more long runs may be 
expected to occur and the total number of runs will be small. The 
hypothesis of randomness may therefore be tested by means of the total 
number of runs, a small number of runs being significant and further 
by means of length of runs, very long runs being significant. 

The sequence of observations for Kj^ values gives the dis- 
tribution of runs up and down as shown in Table 6. For a total number 
of observations of 83 for K, values, the expected number of runs and 
the corresponding variance are given by equations 15 and 16 as 55 and 
14,4 respectively. The difference between the observed and expected 
number of runs is -2. The standard deviation for the total number of 
runs is 3.8, the variance being 14.4. Hence the deviation of the 



55 



TABLE 6 

DISTRIBUTION OF RUNS UP AND DOWN FOR THE 

SEQUENCE OF OBSERVATIONS FOR K x VALUES 

OF THE OHIO RIVER SAMPLES 



Length 
of Run 
i 


Number 
Up 


of Runs Observed 

Both 
Down (x-j) Ry. 


Expected 
E(r|) 


Number 
E(R k ) 


1 


18 


15 


33 


53 


34.7 


55 


2 


5 


8 


13 


20 


16.3 


20.3 


3 


3 


2 


5 


7 


2.9 


4.0 


4 





2 


2 


2 


0.9 


1.1 


> 5 














0.2 


0.2 


Total 


26 


27 


53 


■BIB 


55.0 


*e*» 



observed total number of runs from the expected value is not significant. 
Also very long runs are not present in the sequence of observations. 
Hence the hypothesis that the observations for K 1 values form a random 
sample cannot be rejected. 

Turning Points Test . This test consists in counting the 
number of peaks and troughs in the series (Kendall and Stuart, 1966), 
A "peak" is a value which is greater than the two neighboring values. 
If there are two or more equal values which are greater than their 
predecessor and successor (a rare event in general) they will be regarded 
as defining one peak. Likewise a "trough* 4 is a value which is lower 
than its neighbors. Both peaks and troughs are treated as cases of 
"turning points" and the interval between two turning points is called 
a phase. The expected number of turning points, P, variance inP, V(j»), 
and the expected number of phases Pj of length i for a sequence of n 



>bservations are given respectively as 



56 



E(P) - 2 (n - 2) 



(17) 



V(P) - ^ (16n - 29) 



(18) 



ind 



E(P ) - 2(n - ^- 2)(^ 2 + 3j + 1) 
MP i ; (i + 3)J 



(19) 



For the river K. values, the actual number of turning points 
in the sequence of observations is 52 as against the expected value of 
>4,7. The distribution of "phases'* for the observed Kj values are shown 
In Table 7. The observed values for the total number of turning points 
ind for the "phases" of different lengths are so close to the theoretical 
ralues that the hypothesis of random variations in K. values cannot be 



rejected. 



TABLE 7 

DISTRIBUTION OF "PHASES" FOR THE SEQUENCE 
OF OBSERVATIONS FOR Kj VALUES 
OF THE OHIO RIVER SAMPLES 



Phase 


Number 


of 


Phases 


Number of Phases 


Length 


Observed 


Theoretical 


1 




32 




33.3 


2 




13 




14.5 


3 




5 




4.1 


4 




1 




0.9 


Total 




51 




52.8 



57 



Hypothesis and Hypothesis Testing 

The frequency of occurrence of the observed K^ (20°C) values 
for the Ohio River samples in different class intervals are shown in 
Table 8, The sample mean and standard deviation are respectively 0,173 
per day and 0.066 per day with a coefficient of variation of 38.0 per- 
cent. The histogram for the observed frequency distributions is shown 
in Figure 10 along with a theoretical normal curve having a mean and 
standard deviation of 0.173 per day and 0.066 per day. Figure 11 shows 
these results plotted on a normal probability paper. It is seen that 
all the observed points lie very close to a straight line except at the 
tail ends of the curve fitting where the deviations of the points from 
the straight line are negligibly small* Since the observed frequency for 
K^ values fits very well with the theoretical normal distribution, it is 
hypothesized that K^ values for Ohio River samples at Cincinnati Pool, 
vary randomly according to Gaussian probability with a mean of 0.173 per 
day and a standard deviation of 0.066 per day. This hypothesis is sub- 
jected to further statistical tests as outlined in the following sections, 

TABLE 8 

FREQUENCY DISTRIBUTION OF OBSERVED K L VALUES 
OF THE OHIO RIVER SAMPLES 



Class Interval 


Number 




Percent 


Cumula- 




of Ob- 


Relative 


Relative 


tive 




servations 


Frequency 


Frequency 


Frequency 



0.000 to 0.049 1 

0.050 to 0.099 11 

0.100 to 0.149 20 

0.150 to 0.199 23 

0.200 to 0.249 15 

0.250 to 0.299 11 

0.300 to 0.349 2 



0.012 


1.2 


1.2 


0.133 


13.3 


14.5 


0.241 


24.1 


38.6 


0.277 


27.7 


66.3 


0.181 


18.1 


84.4 


0.133 


13.3 


97.7 


0.023 


2.3 


100.0 



58 






1.0 



0,8 



0.6 



I 

fc 0*4 



0.2 



O.OLa 




0.0 0.03 t.10 0.15 0.20 0.25 0.: 

Deoaygenatim Coefficient, K 1# par day 



FIGURE 10. FREQUENCY PLOTS FOR K t VALUES OF THE OHIO RIVER SAMPLES 



59 



0.01 
0.05 - 

oao _ 

0.20 




99.99 



0.05 0.10 0.15 0.20 0.25 0*30 0.35 0.40 0.45 0.50 
^oxygenation Coefficient, K , per day 



FIGURE 11. CUMULATIVE PROBABILITY DISTRIBUTION OF K, VALUES 
FOR THE OHIO RIVER SAMPLES 



60 

Chi -Square Test . This is a simple test generally used for 
testing whether the data constitute a sample from a population with 
probability density function (p.d.f .) f (x) at the significance level a 
(Ostle, 1964). The range of observed values is divided into a number 
of categories; and the expected number of samples ej, that will fall 
in each category i, as predicted by the p.d.f. f(x), is calculated. 
The observed frequencies nj in each of the categories are compared 
with the expected frequencies by computing the chi -square statistic 
defined by 

r (n, - e.) 2 

X 2 « E h (2°) 

i - 1 

Let f be the number of degrees of freedom, given by f - 
r - 1 - g in which r is the total number of categories, and g is the 
number of population parameters estimated from the sample and used in 
theoretical distribution. If the calculated values of X 2 exceeds the 
value of x a f» the hypothesis that the sample belongs to the distribution 
f(x) is rejected at significance level a. 

The result of chi -square test is shown in Table 9. Merging 
the first and the last of the categories indicated in the table with 
their respective neighboring categories, the resulting number of cate- 
gories is 5 and the number of degrees of freedom is 3. Since x 2 " 1*878 
being less than the critical value x 2 5 3 - 9,35, the hypothesis cannot 
be rejected. 



TABLE 9 



CHI -SQUARE TEST FOR THE GOODNESS OF FIT 
CONCERNING THE HYPOTHESIS FOR OBSERVED K l VALUES 



61 



Class Interval Observed Expected 



0.000 to 0.049 
0.050 to 0.099 
0.100 to 0.149 
0.150 to 0.199 
0.200 to 0.249 
0.250 to 0.299 
0.300 to 0.349 



Frequency Frequency n, - e^ (nj - e.) /ej 
(nj) (ej) 



1 


2.6 






11 


8.5 


0.9 


0.073 


20 


19.0 


1.0 


0.053 


23 


24.6 


1.6 


0.104 


15 


18.3 


3.7 


0.748 


11 


7.8 


3.0 


0.900 


2 


2.2 







2 - 1.878 



Kolmogorov-Smirnov Test . An alternative to the chi -square 
goodness of fit test is provided by the Kolmogorov-Smirnov test (Massey, 
1951; Ostle, 1964). If a population is thought to have some specified 
cumulative distribution function (c.d.f .) F(x), that is, for any 
specified value of x, the value of F(x) is the proportion of individuals 
in the population having measurements less than or equal to x. The 
cumulative step-function of a random sample of n observations is ex- 
pected to be fairly close to this specified distribution function. If 
it is not close enough, it is evident that the hypothetical distribution 
is not the correct one. 

If S n (x) is the observed cumulative step-function of a sample, 
I.e., s n (x) -» k/n, where k is the number of observations less than or 
equal to x, then the sampling distribution of d being the maximum of 
IF(x) - S n (x) I is taken as the statistic for testing the goodness of 
fit. The d statistic should be less than the critical value which de- 
pends on the number of observations on which the hypothesis concerning 



62 

the population distribution is based. Massey (1951) has presented 
proof for the fact that the Kolmogorov-Smirnov test is more powerful 
than the chi -square test and that this test will detect smaller devia- 
tions in cumulative distributions than the chi-square test. 

Results of the d test for the Kj values observed on the Ohio 
River samples are shown in Table 10. From the table it is seen that 
the maximum value for the difference between the theoretical and observed 
cumulative distributions is 0.035. The critical value for the d sta- 
tistic at 5 percent significance level for a total of 83 observations 
is 0.149. Since the observed value for the d statistic is less than 
the critical value, the hypothesis cannot be rejected. The Kolmogorov- 
Smirnov test, applied to individual observations instead of the grouped 
data as presented in Table 10, is also satisfied. 

TABLE 10 

KOLMOGOROV-SMIRNOV TEST FOR THE GOODNESS 

OF FIT CONCERNING THE HYPOTHESIS 

FOR OBSERVED K x VALUES 



Value of 
Variable 


the 
(x) 


S n (x) - 


k 
n 


Normalized 
Variable 


F(x) 


IF( 


x) - S n (x)l 


0.05 




0.012 




-1.864 


0.031 




0.019 


0.10 




0.145 




-1.106 


0.134 




0.011 


0.15 




0.386 




-0.348 


0.368 




0.023 


0.20 




0.663 




+0.409 


0.659 




0.004 


0.25 




0.844 




+1.167 


0.879 




0.035 


0.30 




0.977 




+1.924 


0.973 




0.004 


0.35 




1.000 




+2.682 


0.996 




0.004 



Since the effect of temperature on the deoxygenation coefficient 
K} has been extensively studied and found to be adequately expressed by 
the deterministic equation, Eq. 8, it is assumed that the variations in 
Kj values at any temperature other than 20°C will also be normally 



63 

distributed. Also no attempt was made in this study to verify the 
goodness of fit with other theoretical distributions like lognormal 
distribution, gamma distribution, t and F distributions, etc. However, 
the procedure involved in the verification of the goodness of fit with 
any other theoretical distribution will be the same as that for the 
normal distribution discussed in this work. 

VARIATIONS IN REAERATION COEFFICIENT 
Background 

The rate of reaeration, under constant conditions of tempera- 
ture and turbulence, is directly proportional to the oxygen deficit in 
the water. For many years attempts have been made to derive a method 
which would permit the prediction of reaeration coefficient, relating 
some of the easily obtainable hydraulic characteristics of a stream. 

O'Connor and Dobbins (1956) presented a theoretical derivation 
of the reaeration coefficient in which they attempted to define the rate 
of reaeration in terms of the rate of renewal of the surface film, and 
assumed that the best estimate was given by the ratio of the vertical 
velocity fluctuation and mixing length. In streams in which there is 
pronounced velocity gradient, the following equation for nonisotropic 
turbulence was developed: 

480 D„ - 5 S - 25 

*2 " — J5 

and for isotropic turbulence, In relatively deep channels where there 
is no pronounced velocity gradient: 



64 



127 D °' 5 S ' 25 
K 2 " fl (22) 



where D m is the coefficient of molecular diffusion, S is the channel 
slope and H is the mean depth. Isotropic turbulence was assumed ar- 
bitrarily for Chezy coefficients greater than 17 and nonisotropic 
turbulence for those less than 17. O'Connor and Dobbins (1956) at- 
tempted to verify their equations by comparing predicted values of the 
reaeration coefficients for the Ohio River using the Streeter-P helps 
oxygen sag equation (1925)* They obtained good agreement in spite of 
the fact that errors in the estimation of the parameters such as the 
time of water travel through each reach, the DO concentration, ultimate 
BOD, deoxygenation coefficient, etc., for use in the Streeter-Phelps 
equation, are reflected in the computed value of K£. 

Churchill et al . (1962), using multiple regression techniques, 
determined the coefficients of different equations, formulated by dimen- 
sional analysis, relating observed reaeration rates and various hydraulic 
parameters of river reaches. Corrections for the effects of photosyn- 
thesis wen made in computing reaearation coefficients from observed 
measurements of changes in oxygen concentration in selected stream 
reaches. They concluded that it is possible to predict reaeration rate 
for a river reach from the average velocity, depth, and temperature of 
the water using the equation: 



K 2(20) " 5 ' 026 V ' 969 H" 1 ' 673 (23) 



65 

where V is the mean velocity of flow and H is the mean depth. They 
also suggest that suitable correction should be made for the effects 
of pollution in predicting stream K2 values. Inclusion of other stream 
characteristics such as slope, friction coefficient, and Reynolds 
number did not markedly increase the accuracy of the predicted value. 
A direct correlation between experimentally determined re- 
aeration coefficients in an artificial channel and the corresponding 
values for longitudinal mixing coefficient was demonstrated by Krenkel 
and Orlob (1963). They proposed the empirically derived equation: 

K 2(20) " 1 « 138 x 10 ~ 5 Dl 1,321 H" 2 ' 32 (24) 

where EL is the longitudinal mixing coefficient and H is the depth of 
flow. 

Dobbins (1964) proposed a mathematical model for the re- 
aeration process based on the concept of "an interfacial liquid film" 
which maintains its existence in the statistical sense, that is the 
film is always present but the liquid content of the film is being 
continuously replaced in a random manner by the liquid from the main 
body. This concept led to the formulation of the equation 



K 



i rL t 
- (D^r)* coth qgjp (25) 



in which K L is the liquid film coefficient, r is the average frequency 
with which the stagnant film is replaced and L t is the thickness of 
film. He further developed several equations using postulated constants 
to arrive at the relationship between K^, r, L, and K2 applicable to 



66 



natural streams. His attempts to establish numerical values for the 
hypothesized constants and proportionality factors were somewhat less 
than successful. Thackston and Krenkel (1965) have raised objection 
to the correctness of the theory proposed by Dobbins (1964), questioning 
the validity of the propounded relationships between the constants used 
in the theory, some of which are based solely on judgment. 

Several other attempts to formulate empirical relationships 
for the prediction of K 2 values in rivers have been reported in the 
literature. Owens et al. (1964) proposed an equation based on the data 
efatatsed in several reaches of lowland and Lake District streams in 
England, using multiple regression analysis as: 

K 2(20) " 10 « 09 v °* 73 H" 1 ' 75 (26) 

Also Langbein and Durum (1967) gave the following relationship: 

K 2 - 3.3 V H" 1 * 33 (27) 

In all these prediction equations developed either on theoret- 
ical basis or using regression analysis, considerable scattering of the 
observed values for K 2 are noticed about the prediction line. As has 
been mentioned earlier, in one of the reaches of the Holston River in 
the Tennessee Valley, the observed K 2 value ranged from 0.10 to 1.18 per 
day at 20°C for constant discharge in the river whereas the predicted 
value for the reach was 0.59 per day* Also in another reach of Watauga 
River which is a shallow and rapid stream compared to Holston River, 
observed values for K 2 varied from 2.26 to 8.86 per day whereas the pre- 
dicted value using regression Eq. 23 developed for these rivers was 3.22 



67 

per day. Most of the mathemat leal models developed so far for predicting 
K2 values in rivers use at best the average values for river velocity 
and depth. Since these change from section to section and also within 
the section itself in natural streams, the estimated average values do 
not yield satisfactory results. Also errors in sampling and analyses, 
and in the estimation of mean travel time combined with the uncertainties 
of wind effects, all complicate the prospects of predicting K2 reasonably 
accurately. 

Eckenf elder and O'Connor (1961) indicated that the variations 
in K2 depend upon the hydraulic properties of the particular channel: 
roughness, width and curvature, and that the variations are usually 
defined by a normal distribution. They further suggested that any value 
of the reaeration coefficient within limited statistical ranges could be 
used in oxygen balance calculations. O'Connor (1958) presented data, 
taken from actual stream surveys, showing variations of depths from 
station to station in the Wabash, Clarion and Codorus Rivers. These 
were found to be normally distributed and he discussed the importance 
of considering these variations in determining the required number of 
cross sectional areas to insure for a given probability that the measured 
mean depth for use in prediction equations for K2 will fall within a 
given percent of the true mean. 

Not only the variations in depth affect the evaluation of K 2 
values, but also several other factors discussed earlier come into play. 
It becomes necessary to ascertain the combined effects of all these 
factors, since it is not possible to assess the effects individually in 
most cases. The data collected by the Tennessee Valley Authority (1962) 



68 



on natural streams with constant discharges for each set of observations, 
being the most extensive, will be used in this study to determine the 
extent of variations and the parameters characterizing these variations. 

Modified Regression Equation 

The data, obtained in Tennessee Valley rivers (1962), com- 
prised of 509 Individual observations under 30 different sets of 
experimental conditions. In deriving the regression equations, 
Churchill et al. (1962) used geometric means of observations for 
K2 in each set of experiments. In the knowledge of this author, most 
of the statistical theories concern themselves with the arithmetic 
means and not with geometric means. In the application of statistical 
and probability theories to real phenomena two results play a con- 
spicuous role (Ostle, 1964; Parzen, 1966). These results are known 
as the Law of Large Numbers and Central Limit Theorem, and deal with 
the arithmetic means of observations. Hence a regression analysis of 
the data obtained in Tennessee Valley rivers using arithmetic means 
of group observations was carried out in this study. An abstract of 
published data, arithmetic means of observations, and the residual 
errors in the regression analysis are shown in Appendix E. 

A general linear regression equation may be represented by 

Zq - A Q + k l Z l + AjZj* ... A p Z p (28) 

where Zq is the dependent variable estimated from the independent 
variables Z^, Z2,...,Z . From the historical data, Z 4 , with i - l,...,p 
are known and Z Q * f the actual value of dependent variable Z is also 



69 

known. For the least squares linear regression, the coefficients Aq, 
Ap...,A are so evaluated that the sum of the squared errors 



S E 2 - £ (Z « - Z) 2 (29) 



is minimized. Details of solution are available from standard text 
books on statistics (Ezekiel and Fox, 1959; Ostle, 1964). Z ' - Zq 
is referred to as the residual error, or as the random component, 
and so the least square regression minimises the variance of the 
random component. 

The coefficients A^, with i - l,...,p are known as the net 
multiple regression coefficients of Zq on Zj, The coefficient of 
multiple correlation Rq.1 2 is a measure of tne proportion of 
the variance of Zq that is explained by the multiple linear regression 
equation. It is given by 

P n 

o i *, A i f , 2 1 AZ 0J AZ 1J ] 

R 2 - i * 3 * (30) 

u.i,i,...,P n o 

Z LZ 0j 
J " 1 

In the IBM 7094 system available at the University of Illinois, there 
is a standard program of the Statistical Services Research Unit for 
calculating various statistical estimates of multiple linear regression 
including the regression coefficients, the coefficient of multiple 
correlation, etc. It was used in this study for multiple regression 
analysis of the data for river reaeration coefficients. 



70 



The regression analysis yielded an equation 

K 2(20 ) - 5.827 V ' 924 H- 1 ' 705 (31) 

with a correlation coefficient of 0.917 which is better than the 
reported correlation coefficient of 0.822 (Churchill et al., 1962) 
using geometric means for the group data. The form of the regression 
equation was linearized by taking logarithm of both the sides of the 
equation before applying the general linear equation, Eq. 28. Results 
of the regression analysis are shown in Figure 12. There is consider- 
able deviation of the observed values for K 2 from the predicted values, 

An analysis of the distributions of percent error indicated 
that they are normally distributed with mean zero and a standard 
deviation of 36.8 percent as verified by Kolmogorov-Smirnov test for 
goodness of fit. Percent error was obtained by dividing the residual 
error by the predicted value and multiplying the resulting fraction by 
100. Figure 13 shows the cumulative probability distributions of per- 
cent residual errors in predicting K 2 values, plotted on a normal 
probability paper. The nature of the distribution of errors in the 
prediction of K 2 values is not affected by temperature changes, since 
the basic assumption in any regression analysis is that the errors in 
prediction are normally distributed. 

Since the error distributions are considered random, it is 
hypothesized that K 2 values for Tennessee Valley rivers could be pre- 
dicted by superimposing a random component governed by a normal proba- 
bility distribution with a mean zero and a standard deviation of 36.8 



71 



u 



o 

CM 



I 

u 

I 



CM 



3 

o 

> 



6.0 



5,0 



4.0 



3.0 



2.0 — 



o* 1*0 



0.0 



T 


i i i i 




Sn Table 


l of Churchill et al. / 




(1962) for Experiment / 
Identification 8 / 


- 


— 


9 / 30 


- 


29. 






— 10 


1 Multiple Correlation 


- 


17 / 


'20 Coefficient - 0.917 




/ # 2 
WJV* -6 




- 


7 v^m 


1 1 1 1 





0.0 1.0 



2.0 



3.0 



4.0 



5.0 6.0 



Observed Values of K 2 , per day, at 20°C 



FIGURE 12. OBSERVED VERSUS PREDICTED VALUES OF K, 



72 



g 
o 

9 

2 

r 

u 

m 

i-t 

1 



0.01 

0.05 

0.1 

0.2 

0.5 
1.0 
2.0 

5 
10 

20 
30 
40 
50 
60 
70 

80 

90 
95 

98 
99 

99.8 
99.9 

99.99 



1 


1 


1 1 1 ' 1 1 


_ 







mlml 






- 







— 




— 


— 




— 


— 


iV 


w Mean ■ 0.0 percent — 


— 




N. Standard Deviation - 36.8 percent" 


__ 




\ 


— 




— 


1 


J L. 


1 1 11 1 



-50 -30 -10 +10 +30 +50 +70 +90 
Percent Deviation of Predicted K 2 Values 



+110 



FIGURE 13. PROBABILITY DISTRIBUTION OF PERCENT RESIDUAL 
ERRORS IN PREDICTING K 2 VALUES 



73 



percent, onto a trend component predicted by the regression equation, 
Eq. 31 • Since the standard deviation of the percent error involved 
In the predict! <ns of K2 values is considerable, there seems to be 
little justification in ignoring this random error component altogether 
in evaluating stream reaeration coefficients. 



74 



V, MONTE CARLO METHOD AND SIMULATION MODEL FOR 
STREAM ASSIMILATIVE CAPACITIES 



MONTE CARLO METHOD 
General 

Monte Carlo methods comprise that branch of experimental 
mathematics which is concerned with experiments on random numbers 
(Hammersley and Hands comb, 1964; Hammersley and Morton, 1954; Kahn, 
1956; Meyer, 1954). Problems handled by Monte Carlo methods are 
generally classified into two types; probabilistic or deterministic 
according to whether or not they are directly concerned with the 
behavior and outcome of random processes. In the case of probabilistic 
problems the simplest Monte Carlo approach is to observe random numbers, 
chosen in such a way that they simulate physical random processes of 
the original problem, and to infer the desired solution from the be- 
havior of these random numbers. Monte Carlo methods have found ex- 
tensive use in the fields of nuclear physics and they have been employed 
in other fields of science like chemistry, engineering, biology, medi- 
cine, etc. The significance and the concepts involved in Monte Carlo 
methods are brought out by considering two examples. 

The central problem in designing a nuclear reactor is to 
determine the critical size of the core. Except when the geometry is 
simple or the neutrons have small energy ranges, analytical techniques 
are difficult and recourse is often made to Monte Carlo techniques. A 
free neutron, put into the core, performs a random walk there, colliding 
with fissile nuclei until it either escapes from the core or is absorbed 
by a nucleus. If the former occurs, the neutron continues its random 
walk in the reflector; if the latter occurs, a fission may result 



75 

ejecting several free neutrons, called descendants, each of which 
begins its own random walk. The size of the core is critical when 
the average number of neutrons in it stays constant* Three proba- 
bility distributions determine such a random walk (Hammersley and 
Morton, 1954): 

1. The distance a neutron travels between successive 
collisions is distributed exponentially with a mean called the total 
mean free path (varying from medium to medium)* At a boundary a 
neutron continues, i n the same direction with the same energy, a 
distance with a mean equal to the total mean free path in the new 
medium* 

2* There are assigned probabilities of the possible types 
of collision, namely capture, scattering, or fission and their results 
including energy changes* 

3. The direction a neutron takes after a collision is 
often isotropically distributed, but for collisions with light nuclei 
is peaked in the direction of the incident neutron's path. 
These three distributions depend on the energy of the neutron concerned. 
Knowing them, one can conduct a neutron from collision to collision by 
sampling, recording the position, energy, and direction of its motion 
and its descendants just before each collision* A tally for the number 
of neutrons in the core is kept due to several parent neutrons in the 
core as a function of a so-called "census parameter," namely a quantity 
measuring the current duration of the process (e.g., the number of 
collisions to date, or the total distance traveled, etc*)* This having 



76 



been done for various core sizes, one can estimate the critical size 
by interpolating for a constant tally between the increasing and de- 
creasing tallies. The performance of a nuclear reactor is simulated 
here by choosing random numbers which represent the random motions of 
the neutron in it. In this way, one can experiment with a reactor 
without incurring the cost, in money, time, and safety, of its actual 
physical construction. 

As an example for the application of the Monte Carlo tech- 
niques in the field of biology, if one wishes to study the growth of 
an insect population on the basis of certain assumed vital statistics 
of survival and reproduction, a model could be set up with paper entries 
for the life histories of individual insects. To each such individual, 
random numbers are allotted for its age at the births of its offspring 
and its death; and then treat these and succeeding offsprings likewise. 
Handling the random numbers to match the vital statistics, one gets 
what amounts to a random sample from the population which can be anal- 
yzed just as real experimental data collected in the field. 

Thus it is seen that the Monte Carlo method involves essen- 
tially manipulations of random numbers in the simulated model for 
physical processes and is based on the assumption that in a game of 
chance, the expected outcome can be estimated in principle by averaging 
the results of a large number of plays of the game. Also methods em- 
ploying Monte Carlo techniques have afforded workable solutions in a 
widely divergent area of endeavors. 

There are essentially three steps involved in solving a 
problem using Monte Carlo techniques. They are: 

1. Choosing or analogizing the probability process. 



77 

2. Generating sample values of the random variables* 

3. Use of variance reducing techniques in the Monte Carlo 
predictions. 

The first of these will be considered in a subsequent section of this 
chapter in order to maintain the continuity of the discussion on the 
techniques of Monte Carlo methods. Attention will be focused now on 
the methods generally used in developing sample values of the random 
variables. 

Random Sampling from Specified Distributions 

Sources of Random Numbers . Mechanical sampling devices like 
disks and roulette wheels have been used for generating random numbers. 
Since the wheel spinning methods are tedious and cumbersome, they have 
given way to tables of random numbers and use of computing machines for 
developing a series of random numbers. 

There are several tables of random numbers (The Rand Corpor- 
ation, 1955; Tippet, 1927; Wold, 1954) which have been obtained by 
performing permutations on the high order digits of mathematical 
tables, by compiling from lottery drawings, etc., or from some chance 
devices. They have been subjected to several statistical tests and 
are found acceptable for general sampling use. Usually these tables 
have uniformly distributed decimal digits, though there are some that 
have Gaussian or normal deviates. In such tables, the first number 
to be used is generally arbitrarily selected, perhaps by opening a 
page at random, and with eyes closed, touching a number in the table 
with a finger. The rest of the numbers are usually chosen in sequence. 
Random number tables are very convenient for generating a fairly small 
sequence. 



78 



Random Number Generators . Several mechanical or electronic 
systems have been developed for generating a sequence of random numbers 
or variables (Tocher, 1963). The electronic machines are based on the 
principle of converting a source of random noise into a train of pulses 
which are used to drive a cyclic counter. The drive is interrupted 
at fixed intervals of time, and the successive status of the counter 
gives the successive random numbers. Machines that use analog systems 
in which the random input is self generated are known as analog ran- 
domisers. A detailed discussion of the random number generators is 
given by Tocher (1963). 

Development of Pseudo-Random Numbers . Though random number 
tables are available as punched cards, with increasing use of digital 
computers, mathematical methods for generating * 'pseudo-random" numbers 
within the computing machines have been developed in order to eliminate 
the need for extensive input of random numbers (Kahn, 1956; Meyer, 1954; 
Tocher, 1954), Also when it is desired to repeat the calculations for 
checking purposes, the random numbers used in the initial calculations 
must be available for the later calculations, and hence pseudo-random 
numbers are useful for generating a reproducible random number sequence. 
Though the reproducibility implies the possibility of prediction of the 
sequence, and hence its nonrandomness, the advantages of reproducibility 
make it desirable to investigate the possibility of generating approxi- 
mately random sequences by deterministic processes. Pseudo-random 
numbers may be defined as reproducible sequence of numbers developed 
by using a deterministic process that behaves like a random sequence 



79 

when subjected to certain standard statistical tests. Methods for 
generating pseudo-random numbers include the mid-square method, the 
mid-product method, the congruence methods and other miscellaneous 
methods* Only the multiplicative congruence method which is used in 
this study is discussed below briefly. 

Multiplicative Congruence Method, --One of the characteristics 
of a pseudo-random sequence is that the sequence is cyclic (Meyer, 
1954; Tocher, 1954). Thus attempts have been made to generate cycles 
of maximum length so that a long nonrepetitive sequence is produced. 
Lehmer suggested the use of the theory of numbers for developing such 
long cyclic sequences. The theory of congruences deals with such 
sequences. In this theory, a number Xj, is said to be congruent with 
X2 modulo M if x 2 - Xj is divisible by M. 

Consider a first order recursion equation of the form 

x n + 1 - k^ (mod M) (32) 

where k and M are initially arbitrarily chosen. A model given by 
Eq, 32 is known as a congruential multiplicative model. Then for 
this model, 



x„ » k n x (mod M) (33) 

n o 



Since, at the end of a cycle of length n, 



x - x -» kV (mod M) (34) 

n o o 



the value of k n should satisfy the equation 



80 

k n S l (mod M) (35) 

For use in digital computers in which numbers are represented in the 
binary system, it is preferable to assume M - 2P where p is an Integer, 
to facilitate division. It can be shown (Tocher, 1954) that for M - 2P, 
the maximum possible cycle is of length 2 P , and the only value of k 
giving this cycle must satisfy the condition 

k - +3 (mod 8) (36) 



and x 



is to be odd. A suitable choice of k is any odd power of 5 since 
5 2 <1 + 1 . -3 (mod 8); with q » 1, 2,... (37) 



Also p is to be chosen as the largest number of binary digits shown in 
the computer. After multiplication by k, the least significant p 
digits of the double length product are used as ^ + j. 

A subroutine available at the University of Illinois for 
generating random numbers uses k ■ 5", p - 36, and the initial value 
of x is set to 2 35 - 1. It is also possible to preset this initial 
value to any other chosen value with the restriction that the chosen 
number should be odd. The random sequence when read as a binary 
fraction gives the uniformly distributed random number sequence. The 
subroutine has been tested for the randomness of the sequence. 

Transformation of Random Numbers 

In many problems a random sample from a specified or known 
distribution, which is generally different from the uniform distribution, 
is required. A random sample with any given distribution can be obtained 



81 

(Kahn, 1956; Tocher, 1954) from a given sequence of uniformly distributed 
random numbers by using the inverse probability integral transformation. 
This is based on the basic fact that for any distribution, the cumulative 
probability distribution function has a rectangular distribution, uni- 
form over (0,1). Let x be the random variable with a cumulative dis- 
tribution function, denoted by c.d.f., and designated by F(x). The 
random variable Y • F(x) is given by 



x 

Y - F(x) - J f(x)dx (38) 



where f(x) is the probability density function denoted by p.d.f. 

Consider the probability distribution of Y. Let the p.d.f. 
and c.d.f. of Y be g(y) and G(y) respectively. Then, 

G(y) - Pr {Y < y} - Pr [X < x} - F(x) - Y 

f or < Y < 1 (39) 

and 

g(y) - d |£^ - 1 for < Y < 1 (40) 

dy — — 

This is the probability density function for the uniform (0,1) distribu- 
tion. By taking a uniform (0,1) sample of Y, and taking the Inverse 
x - F -1 (y) of the probability integral transformation Y - F(x), a sample 
of X is developed. By repeating this for as many samples as required, a 
sequence of random numbers having any given distribution can be developed 
from a sequence of uniform (0,1) random numbers. 



82 



The process of Inverse trans format ion can be done by several 
methods (Tocher, 1954). A relatively simple method for the inverse 
probability integral transformation is the use of interpolation from a 
table having corresponding values of x and y, and the use of a suitable 
degree of polynomial for interpolation. This method, though unsophis- 
ticated, is fairly simple and was used in this study to transform the 
c.d.f. to the random number. 

Variance Reducing Techniques 

The methods which are most often used to reduce variance in 
Monte Carlo problems are straightforward sampling, systematic sampling, 
stratified sampling, use of expected values, correlation, and Russian 
roulette and splitting (Kahn, 1956; Meyer, 1954). The methods which 
can be used to reduce variance are often dependent upon the probability 
model and in some cases on the techniques used to generate values of 
the random variables. Kahn (1956) has given an excellent discussion 
of the general nature of the techniques with their applications to 
evaluation of integrals. The first of these which has found extensive 
use in engineering problems, and which is used in this study is dis- 
cussed below. 

Straightforward Sampling . This technique is based on the 
premise that one way or reducing uncertainty in an answer is to base 
it upon more observations (Hammersley and Hands comb, 1964; Kahn, 1956; 
Tocher, 1954). Broadly speaking there is a square law relationship 
between the error in an answer and the requisite number of samples. 
If a sequence of n random variables x^, X2,«..,x n are picked from the 
P*d.f. f (x) and If a random variable s n defined by the equation 



83 



n 

*n - J 2 ■<*!> < 41 > 

i « 1 



and the integral 



J z(x)f(x)dx (42) 



exist in the ordinary sense, z n will almost always approach z as a limit 
as n approaches ». 

The integral Eq. 42 is called the expected value of the 
function z(x), and z is called an estimate of z. If z , the expected 
value of z 2 (x) also exists, an estimate can be made about the amount 
that z n deviates from z for large n. Denoting the variance of z(x) by 
<r, it is given by the equation 



a 2 m (z - ») 2 (43) 

- J (SB - Z) 2 f (X)dx 

- Jz 2 f(x)dx - 2z j zf(x)dx 

+ z 2 J f (x)dx 

m Z 2 - 2Z 2 + Z 2 

- z 7 - z 2 (43a) 

Applying the Central Limit Theorem (Parzen, 1966), for large n 
the probability that the event z -6 < z n < z + 6 occurs is asymptotically 
independent of the exact nature of z(x) or f(x) but depends only on n 



84 



and cr» It can be deduced from the Central Limit Theorem that 



, *& 2 

p r { S„ < z + 8} - -±j ? exp (- 5_)dx (64) 



/* 1 "^-t 



and that o^s » the variance of the estimator z_ of z is given by the 
n 



equation 



A - £ < 45 > 



z n 



or 



n /h 



Thus, it is seen that there is a square law relationship between the 
error in an answer and the requisite number of observations. To 
reduce the error tenfold calls for a hundred fold increase in the 
observations, and so on. 

Estimation of Sample Size . In any simulation, it is necessary 
to know approximately the sample size required before sampling is 
started in order that the size is neither too large to make it very 
expensive compared to the information gained, nor too small to be 
reliable. When the required precision and confidence levels for the 
reliability of prediction are given, the sample size can be determined 
by the following approach (Flagle, et al., 1960; Massey, 1951). 



85 

Let 3 and a percent be the error and significance levels for 
Che reliability of predicted values in a simulation model. Within the 
same confidence interval, the precision could be increased only by in- 
creasing the sample size. Since the standard deviation and consequently 
the tolerance diminishes as the square root of the sample size, to in- 
crease the precision by a factor of two, the sample sise would have to 
be quadrupled. If ^f n ) is the critical value of the *d' statistic in 
the Kolmogorov-Smirnov test for goodness of fit at a signficiance 
level of a, which is defined as the maximum absolute difference between 
sample and population cumulative distributions, then the required 
sample size is given by the equation 

c 
<k(n) - /n" " & < 46 > 

where c a is the constant defining the critical value of the d statistic 
in the Kolmogorov-Smirnov test for samples of size greater than 35. 
Knowing the values of a and 3, it is then possible to pick out the 
value of Cq^ from statistical tables (Massey, 1951; Ostle, 1964) and 
hence the value of n, being the requisite number of samples to meet 
the stipulated criteria could be computed. 

SIMULATION IN WATER RESOURCES SYSTEMS 

Simulation means to duplicate the essence of the system or 
activity without actually attaining reality itself. Simulation has 
been used traditionally in engineering. The use of conceptual system 
models, scale models, analogs, and laboratory experimentation are but 
some of the general simulation techniques used in engineering. However, 



86 



this study deals with the simulation of processes involved in the 
prediction of a stream's waste assimilative capacities using a 
digital computer* The approximation, complexity, accuracy and cost of 
simulation are somewhere between those of mathematical analysis and 
prototype testing. Simulation is especially adaptable to the study 
of complex systems and in a way, it may be considered as a numerical 
method for the solution of complicated probabilistic or stochastic 
processes* It is used to circumvent the difficulties of duplication 
of environment, of mathematical formulation, of lack of analytic 
solution techniques, or of experimental impossibilities* 

Analog and digital computers are frequently used for pre- 
dicting the responses of certain water quality parameters in surface 
water bodies like rivers, lakes, estuaries, etc* With the advent of 
modern electronic computers, more and more efforts are being directed 
toward water resources planning on a regional basis, with optimization 
of water and waste water treatment processes taken as a whole instead 
of treating these two as separate entities as has been done in the past. 
Since water resources systems are asually complex with multiple alter- 
natives for design, allocation, and operation, simulation seems to be 
a very useful tool in water resources system design* 

When the mathematical model for a process has been decided 
upon, various elements of the process can be represented in the computer 
so that the outputs of one part of the system constitute the inputs 
into one or more other elements* A complex system may thus be represented 
by a series of relatively simple elements. In such a system, it is 



87 



possible to vary the parameters of the system to Investigate some 
specific aspects of the system, and when necessary, to adopt systematic 
or trial and error procedures for the optimization of the system. 
Because of these facilities, digital and analog computers have been 
used extensively for the simulation of dynamic physical systems. 

This study deals with the simulation of a river system using 
a digital computer with specific reference to the simulation of the 
phenomenon of a stream's assimilative capacity for the organic waste 
discharged into it without creating nuisance conditions in the stream. 
This differs from analog models in that the generated data are used for 
the simulation of the process facilitating the study of the probabil- 
istic aspects of the process as a whole. Generated data, as used here, 
will refer to generating random values for variables in the process 
under consideration on the basis of statistical models for the vari- 
ables, which could not be distinguished from the real or historical 
data by means of the usual statistical tests of significance. The gen- 
erated data did not occur in the past, nor will they occur in the 
future; but based on certain statistical considerations, it could have 
been the real record. Regarding validity of simulation Flagle et al. 
(1960) state: 

•••It must be remembered that simulation yields only 
an empirical form of knowledge, fraught with the 
danger that the stochastic model is not truly repre- 
sentative. In this sense it is admittedly inferior 
to mathematical techniques which yield functional 
relationship between the variables. Ultimately we hope 
that the more slowly developing mathematical capa- 
bilities will make simulation as we practice it, un- 
necessary. Until that time comes, however, simulation 
techniques stand as powerful aides to the operations 
analyst who must produce useful results. 



88 



MODEL FOR STREAM ASSIMILATIVE CAPACITY 
Factors Affecting DO and BOD 

Since Streeter and Phelps (1925) propounded their theory 
defining two mechanisms which affect the DO and BOD relationship 
namely, the BOD removal due to bacterial oxidation and oxygen addition 
due to atmospheric reaeration, several other factors affecting DO-BOD 
relationship have been postulated (Camp, 1963; Dobbins, 1964; Gannon, 
1963; Thomas, 1948), Some of the major factors which affect the as- 
similative capacity of a stream in general can be summarized as follows: 

1, Removal of BOD by bacterial oxidation of both carbon- 
aceous and nitrogenous matter* 

2. Removal of BOD by sedimentation or adsorption. 

3* Addition of BOD along a stretch by the scour of bottom 
deposits or by the diffusion of partly decomposed organic matter from the 
benthal layer into the overlying water. 

4. Addition of BOD along the stretch by local run off. 

5. Biological removal and accumulation of BOD by fixed 
plants and algae. 

6. Addition of oxygen from the atmosphere. 

7. Addition of oxygen by the photosynthetic action of 
plankton and fixed plants. 

8. Removal of oxygen by the respiration of plankton and 
fixed plants. 

Thomas (1948) introduced a rate constant K3, with units of per day, as 
a means of accounting for the removal or addition of BOD by deposition 
or resuspension. The net rate was assumed to be proportional to K3L, a 



89 

negative value of K~ indicating deposition and a positive value 
indicating resus pens ion. 

In considering the stabilization of organic matter in 
natural streams, distinction should be made between those factors 
which affect its removal without necessarily utilising oxygen and 
those which simultaneously utilize oxygen while removing the organic 
matter* In the first category are such phenomena as sedimentation, 
scour* and biological accumulation and the second category includes 
various forms of chemical and biological oxidation. Generally when a 
waste is discharged to a stream* it is extremely difficult to isolate 
the effects due to each of these factors individually* O'Connor (1967) 
introduced the parameter K r * with units of per day, to define the total 
rate of removal of BOD in the river* It is determined from a series 
of measurements of the BOD at a number of stations downstream from the 
outfall* Since the rate of removal of organic matter is not necessarily 
equal to the rate at which oxygen is utilized, the difference between 
the BOD removal rate and oxygen utilization, if such a difference 
exists, may be attributed to sedimentation, scour, benthal demand, and 
biological accumulation* If none of these factors is present or sig- 
nificant, it is usually assumed that the rate at which the organic 
matter is removed is equal to the rate at which the dissolved oxygen 
is utilized (O'Connor, 1967), in which case K x - K r . 

The sources of oxygen in a river stretch are the amount of 
DO in the Incoming flow, that due to natural or artificial reaeration 
and that due to the photosynthetic addition. The rate of reaeration 
is proportional to the dissolved oxygen deficit. The photosynthetic 



90 



source depends upon many factors such as sunlight, temperature, mass 
of algae and nutrients. If the effect of these factors is included in 
the term P fct , representing the overall rate at which oxygen is released 
by photosynthesis and if the photosynthetic rate if assumed to vary as 
the sunlight, reaching a peak at noon and a zero value at sunrise and 
sunset, then this source could be defined by a periodic function 
(O'Connor, 1967) as: 



P„ sin St» when < t* < p 

V - { D ^ - - (*7) 

when 1 - p < t' < 1 



where p is the period of oscillation of the periodic function, P m 
is the amplitude of the periodic function and t' is the time of flow 
in days of a mass of water since sunrise after the mass of water has 
entered the head end of a river reach considered. If the period p is 
twelve hours or half a day, the periodic function shown in Eq, 47 
reduces to 



P sin 2nt* when < t« < 1/2 
P t , - { (47a) 

when 1/2 < t» < 1 



This periodic function can be described by a Fourier series as 
defined below: 



00 00 

P t « - a + T a^ cos nt 1 + £ b n sin nt* (48) 
n - 1 n - 1 



91 



The coefficients of the Fourier series can be evaluated using 
Euler's formula (Kreyzig, 1962) giving 



P m 

*o - T < 49 > 



a n " fn ^^tS^^ " W 4 I £* t * > <50) 



When n is odd 



a„ - (50a) 

n 



and when n is even 

2P 



m 



Also 



(n - l)(n + 1)tt 



p m 



(50b) 



b! - Y (51) 



and 



b_ - for n - 2, 3,... (51a) 



Hence 



P P 2P 

P t , - — + Y sin 2 TTt« - — (I cos 4TTt» + Ij Snt* + ...) (52) 

Thus it is seen that the oxygen addition due to photosynthesis can be 
represented by the first three terms of Eq. 52 with good approximation. 
This approach of expressing quantitatively the diurnal variations in 
phot ©synthetic activity is decidedly superior to averaging over the 



92 

entire period including the nighttime flow conditions as was done 
by Camp (1963) and Dobbins (1964), 

Bauat ions for BOD and DO Profiles 

The equations for the BOD and DO profiles along a river 
stretch are based on the following assumptions: 

1. The stream flow is steady and uniform, 

2. Removal of BOD by both bacterial oxidation, sedimentation 
or adsorption or both and biological accumulation are first order re- 
actions, the rates of removal being proportional to the amount present. 

3. The removal of oxygen by benthal demand, by plant 
respiration and the addition of BOD from the benthal layer are uniform 
along the stretch. 

4. Diurnal variations of oxygen addition due to photosyn- 
thesis can be represented by a sine function. 

5. The rate coefficients affecting the oxidation of organic 
matter and atmospheric reaeration can be taken as random variables and 
these could be defined by suitable statistical models. 

6. The BOD and DO are uniformly distributed over each cross 
section so that the equations can be written in the one-dimensional 
form. 

Under the foregoing assumptions, the differential equation 
for the BOD profile is given by 



f - - «i + K 3 * V L (53) 



93 



where K fe is the coefficient for determining the rate of BOD removal 
due to biological accumulation. The differential equation for the 
oxygen profile is given by 

~ - K x h - K 2 D - P ffl (~ + | sin 2-nt' - ^ cos 4TTt') + R + D b (54) 

where R is the rate of oxygen removal by algal respiration and IX It the 
benthal demand both with the units of mg/(l)(day). When these two 
equations, Eqs. 53 and 54, are solved simultaneously with known initial 
conditions for the BOD and DO at the upper end of a river section, it 
is possible to predict the state of DO at any point within the reach. 

Application to the Problem under Study 

Generated data for the values of K., K 2 and DO found necessary 
for the probabilistic analysis of the process under investigation were 
developed by using Monte Carlo methods and simulation techniques. The 
statistical models as developed in this study for K. and K 2 were used 
in generating data applicable to the Cincinnati Pool of the Ohio River 
prior to the construction of Markland Dam. Parameters pertaining to 
other mechanisms namely photosynthesis, respiration, etc., are evalu- 
ated from the published Ohio River Survey data (USPHS, 1960). These 
parameters were taken at their average values and the expected and 
medal values for DO as predicted by the probabilistic model for known 
initial conditions of BOD and DO deficit were compared with the actual 
DO concentrations observed in the survey. 



94 



Using a subroutine available at the Digital Computer Labora- 
tory at the University of Illinois, two sets of uniformly distributed 
random number sequences were generated and transformed to correspond 
to normally distributed random variables with given characteristics. 
The first set of generated values was used to evaluate K^ and the 
second K 2 , In the case of K 1# the generated normally distributed 
values with parameters matching the historical values themselves con- 
stitute the simulated values for K, . In the case of K 2 , the generated 
values constitute the random components of K-> and when these are added 
to the trend component as given by the regression equation, Eq. 31, 
the random variable K 2 is obtained* The generated data were tested to 
see whether the sample could be distinguished from the theoretical 
model using the Kolmogorov-Smirnov test. The generated values for K^ 
and K 2 along with other parameters as determined in the Survey (USPHS, 
1960) were used in the simulation model to obtain probabilistic estimates 
of DO at the desired sections in the river reach considered. 

It is appropriate to quote what Flagle et al. (1960) had to 

say on probabilistic approaches for solving problems which do not 

strictly conform to deterministic rules: 

...as soon as human participation in the system operation 
occurs it is necessary to introduce arguments of proba- 
bility and calculate, not how the system will certainly 
and invariably perform, but only how it will perform a 
certain fraction of the time it is stated. Sometimes, 
as in the case of the atomic bomb already mentioned, 
statistical methods must be used even for a mechanical 
system when the laws of physics that apply (like those 
for radio-activity) are inherently statistical in nature. 

This study is limited to the development and subsequent 

testing of the generated K^ and K 2 data and the analysis of some of 



95 

the probabilistic characteristics of the DO response in stream as- 
similative capacity determinations* Future studies may deal with 
the simulation of systems of waste treatment facilities, and the 
combination of water and waste water treatment facilities for 
determination of the optimal design. 



96 



VI. APPLICATION OF MONTE CARLO METHOD 
TO THE OHIO RIVER SURVEY DATA 



CHOICE OF PUBLISHED RIVER SURVEY DATA 

The primary objective of this study is to develop a method 
for estimating the stream's waste assimilative capacity taking into 
account the variability of the reaction coefficients affecting the 
DO-BOD relationship; the need for which has long been felt. In order 
to express quantitatively the variability in these coefficients 
using probability measure, extensive data are needed and published 
data for five river surveys (Gannon, 1963; The Resources Agency of 
California, 1962; TVA, 1962; USPHS, 1960; USPHS, 1963) were examined 
to study in detail the probabilistic variations in these coefficients 
and to verify the observed data of dissolved oxygen with the predicted 
values using Monte Carlo methods. Though the Sacramento River survey 
(The Resources Agency of California, 1962) was extensive, encompassing 
information on physical, chemical, and biological aspects, photosyn- 
thesis, respiration, etc., the number of observations made for long- 
term BOD progression below a single major waste discharge was far less 
than that made in the Ohio River survey (USPHS, 1960), which was other- 
wise comparable to Sacramento River survey in various other aspects 
observed. The long-term BOD experiment results published by Gannon 
(1963) for the Clinton and Tittabawassee Rivers were carried out at 
different conditions of incubation temperature, mixing, dilution and 
nitrification inhibition agents; whereas in the Ohio River survey tests 
for the BOD progression river samples were all carried out according 
to the procedures stipulated in Standard Methods (APHA, 1955). Also 



97 

no attempts were made in this survey (Gannon, 1963) to evaluate the 
effect of photosynthesis due to aquatic vegetation, even though the 
author concluded that the abundance of aquatic vegetation was respon- 
sible for a manifold increase in the river BOD removal rate compared 
to the BOD removal rate observed in bottle experiments. The report 
on the Illinois River System (USPHS, 1963) did not include details of 
the BOD progression data and the number of experiments carried out in 
this study to evaluate K, values was too few to make reliable statis- 
tical inference. Hence it was observed that the data published for 
the Ohio River-Cincinnati Pool survey (USPHS, 1960) which included 
long-term BOD results on river samples, dark and light bottle tests 
for determining photosynthesis and respiration of phy top lank tons, 
benthal deposits, hydraulic characteristics of the river, etc*, are 
best suited for this study. 

In any river survey for determining the pollution assimilative 
capacity, it is necessary to determine the river reaeration coefficient, 
K2, independently. In order to estimate the random values of K2 for 
the Ohio River-Cincinnati Pool reach, it is assumed that the regression 
equation for the trend component and the probability distribution 
function Cor the random component of K2 as developed for Tennessee 
Valley rivers are applicable to the Ohio River reach under consideration. 

VALUES OF THE PARAMETERS USED IN THE MODEL 
Deoxygenation Coefficient 

With the advent and increased use of secondary treatment 
facilities for waste disposal, the importance of considering oxygen 
demand due to nitrification has been recognized. O'Connor (1967), 



98 

and Stratton and McCarty (1967) proposed mathematical models to account 
for the nitrogenous oxygen demand in oxygen balance studies in rivers. 
As indicated in Chapter IV, in the case of the Ohio River study, neither 
the treatment plant effluents nor the river samples obtained below the 
waste outfalls showed any significant nitrification for about 10 days. 
Though the mathematical model proposed to be used for the Ohio River 
does not contain terms to account for nitrification, conceptually it 
does not present any difficulty to include terms to account for the 
oxygen consumption due to nitrification for situations where this ef- 
fect is significant. Thus, the only factor which affects BOD removal 
with concomitant removal of oxygen is that due to the biological oxi- 
dation of organic matter. The details of the analysis of BOD progression 
data obtained on Ohio River-Cincinnati Pool samples were presented in 
Chapter IV, The variations of K^ values for these river samples were 
found to be random and adequately defined by a statistical model) namely, 
that the Kj^ values for river samples were normally distributed with a 
mean of 0,173 per day and a standard deviation of 0.066 per day having 
a coefficient of variation of 38 percent. This statistical model for 
Kj is adapted for simulation studies using the Monte Carlo method. 

Effect of Sludge Deposits 

One of the greatest difficulties in stream surveys for 
estimating assimilative capacities is to evaluate the values of K-j, 
K5, and D fe individually. To overcome the difficulty of estimating K 3 
and K b values Velz and Gannon (1962), Gannon (1963), O'Connor (1967), 
and others advocate the use of the term K r which defines the coefficient 



99 



for the BOD removal rate due to the lumped effects of K^, K3, K., 
and other unaccounted causes. In the absence of significant sludge 
deposits and fixed vegetation, the values of the rate coefficients K 3 
and K b can be taken as zero leaving only the oxidation of organic 
matter as the significant operative mechanism responsible for the 
removal of BOD from the system. Bottom samples collected during the 
1957 study of the Ohio River-Cincinnati Pool (USPHS, 1960) revealed 
that there rat no significant sludge deposits in the reaches in- 
vestigated to warrant recognition of this as an important factor in 
the oxygen balance studies. This is further substantiated by the 
experience of several investigators (Nejedly, 1966; Thomas, 1948; 
Velz, 1958), that the problem of sludge deposits is greatly allevi- 
ated if settleable solids are removed from the raw waste prior to 
discharge into rivers. Since all the wastes emanating from Cincinnati, 
Ohio received primary treatment, and since the survey report indicates 
that the sludge deposits were insignificant, all the mechanisms of BOD 
and DO removal governed by sludge deposits can be treated as insig- 
nificant and consequently K~ can be taken as zero. 

In deep rivers such as the Ohio River, BOD removal due to 
extraction and accumulation of fixed plants is insignificant (Gannon, 
1963; Heukelekian, 1967). It is relevant here to quote what Gannon 
(1963) has to say on river BOD removal mechanisms: 

Not all rivers have high BOD removal and many have 
rates which closely parallel the rates determined in 
the laboratory bottle experiments. This appears to 
be particularly true for the large rivers, where there 
is relatively less contact with river bottom and sides 
and where there is no gross dispersed type of growth. 



100 

Thus, it is seen that the only significant mechanism of any consequence 
affecting the BOD removal in the Ohio River-Cincinnati Pool area is 
that due to the biological oxidation of organic matter. Also con- 
sequent to the absence of significant sludge deposit, the benthal 
demand for oxygen is negligible, 

Reaeration Coefficient 

Churchill et al. (1962) suggest that when the regression 
equation, Eq. 23, is applied to polluted streams, the basic reaeration 
rates should be modified on a percentage basis as indicated by rela- 
tive reaeration rates determined in the laboratory by experiments for 
the polluted water and for water samples obtained upstream of waste 
discharge. It is generally considered that the constituents in 
waste waters tend to reduce reaeration rates. Downing et al. (1957), 
and Downing and Truesdale (1955) conducted experiments to determine 
the effects of several contaminants and mixtures of contaminants on 
the exchange coefficient for oxygen in water agitated at different 
rates in laboratory absorption vessels. They concluded that house- 
hold detergents reduced exchange coefficients in clean water by 
amounts which depended on their initial concentration and rate of 
agitation. They further concluded that the effects of settled sewage 
in the absence of any added anionic detergent were lower than when 
these materials were present. The reported percent reduction in K2 
values, under laboratory experimental conditions due to municipal 
wastes varies from 10 to 30 depending on the type and concentration 



101 

of the constituents and the rate of agitation (O'Connor, 1958; 
Downing et al,, 1957; Poon and Campbell, 1967)* Poon and Campbell 
(1967) found that at low concentrations of suspended solids in tap 
water, the transfer rates were enhanced to the extent of 10 to 30 
percent* Since the characteristics and concentrations of waste 
discharges are subject to considerable fluctuations, no attempt has 
been made to assign a numerical value for the possible effect of 
the pollutants on the mean river reaeration coefficient as predicted 
by the regression equation used in this study. Since the reaeration 
coefficient is treated as a random variable comprising of trend and 
random components, the error due to neglecting the effects of waste 
constituents which have compensating effects on the reaeration coef- 
ficient is considered negligible. 

The values of velocity of flow, and depth used for deter- 
mining the trend components of K2 are shown in Table 11, for six 
different cases in which observations were made at two different 
sections of the river reach under consideration. In these cases, 
the downstream samples were obtained presumably after a time lag equal 
to the flow-through time from the upstream section of the reach with 
the result that the same body of water had been sampled as it flowed 
down the river. The percent deviation of actual K2 value from the 
predicted value was taken to be normally distributed with mean zero 
and standard deviation of 36,8 percent. The order of magnitude of the 
residual error distribution of K~ values applicable to the Ohio River 
study has been assumed to be the same as that for the Tennessee Valley 
rivers. 



103 



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103 



Photosynthesis and Respiration 

Values for these parameters computed from the dark and light 
bottle observations reported for the Ohio River (USPHS, 1960) are 
shown in Table 11, Since photosynthesis and respiration of phyto- 
plankton are greatly dependent on teaperature and sunlight intensity, 
it is likely that these parameters, over a long period of observation 
will tend to show a trend in their values. It is more realistic to 
consider these parameters as variables with trend and random components 
since the factors which affect these parameters fluctuate having a 
trend with diurnal variations, Siace the scope of the Ohio River sur- 
vey light and dark bottle studies was limited to 12 days spread over 
two months, there is not enough information to discern these trends. 
As the maximum values for phot ©synthetic oxygen addition are assumed 
to occur around mid-day, the number of such observations made during 
the survey Is inadequate to formulate and test any statistical model 
to characterize these parameters. Only arithmetic average values of 
these parameters obtained from the observations for different periods 
are used in this study. It is conceptually feasible to consider these 
two parameters also as random variables provided enough data could be 
collected to establish the trend and random components. 

SAMPLE SIZE IN SIMULATION STUDIES OF THE OHIO RIVER 

It was indicated earlier that the required sample size in 
simulation studies could be determined using Eq. 46, if the desired 
precision and confidence levels are known. Flagle (1960) suggests 
values of 10 percent and 95 percent for error and confidence levels 
respectively. Referring to Table 1 of Massey (1951) for the critical 



104 



values of c^(n) of the maximum absolute difference between sample and 
population cumulative distributions, the critical value of 'd' statistic 
in the Kolmogorov-Smirnov test for 95 percent confidence level is given 



"0.05 " '-f- 
/n 



Thus, substituting the values for c^ and 3, being 1,36 and 10 percent 
respectively in Eq. 46, and solving for n, the required sample size 
is indicated as 185 or approximately 200. 

Hence In order to have confidence level of 95 percent that 
the information on required distribution is not different from the actual 
value by more than 10 percent, the sample size is to be 200 . 

RESULTS OF THE MONTE CARLO METHOD APPLIED TO THE OHIO RIVER 

A flow diagram for the computer simulation studies using the 
Monte Carlo method following the general procedure enumerated in Chapter 
V is shown in Appendix C. Two separate sets, each having 200 random 
numbers with uniform distribution (0,1) were generaged using a sub- 
routine available with the University of Illinois Digital Computer 
Laboratory, The first set of random numbers was transformed to corres- 
pond to K. values with a mean value of 0.173 per day and standard 
deviation of 0.066 per day. The second set of random numbers was 
transformed to correspond to the random variations in percent error of 
predicted K 2 values with a mean value of zero and standard deviation 
of 38,6 percent. 



105 



It is likely that some extremely and unreasonably high or 
low values will be generated by the Monte Carlo method and it becomes 
necessary to ignore such values (Montgomery and Lynn, 1964) or these 
have to be corrected on a reasonable basis (Ramaseshan, 1964), In 
this study the transformed variables with values beyond twice the 
deviation from the stipulated mean were assigned mean values. The 
generated random variates were tested for goodness of fit with the 
assumed distributions using the Kolmogorov-Smirnov test. The distri- 
butions of the generated values were found to satisfy the test. 

With known conditions for velocity and depth of flow, the 
trend component of K2 could be evaluated using Eq. 31, Knowing the 
mean K2 values for the given conditions of flow, the random variations 
in K2 could be computed from the generated random variations in percent 
error. These generated random values for the variations in K2 values 
were added algebraically to the mean value to obtain the random values 
of K 2 , 

The river system for determining the assimilative capacity 
was simulated using Eqs, 53 and 54 in which K3, K,, and D. were taken 
as zero for reasons discussed earlier. Generated values of Kj, K2, 
along with mean values for P m and R indicated in Table 11 were used in 
determining the response of the system. The values for DO deficit 
predicted by Monte Carlo techniques were verified with observed values 
in six different cases in which the same body of river water had been 
sampled at two different sections downstream of all the major waste 
discharges emanating from Cincinnati, Ohio, In each of these cases, 

the process was simulated with known initial conditions and the state 

i 

j 



106 

of dissolved oxygen in terms of oxygen deficit was predicted. The 
initial conditions of ultimate BOD, dissolved oxygen, temperature, 
and the final conditions of dissolved oxygen and tejiperature along 
with the flow-through time between the two sections are presented in 
Table 12, for each of the six cases considered. 

Two hundred values for dissolved oxygen deficit values were 
generated for each case study solving the differential equations using 
a subroutine in the University of Illinois Digital Computer Laboratory. 
The subroutine employs fourth order Runge-Kutta method for solving dif- 
ferential equations, the details of which can be found in standard 
textbooks on numerical analysis (Fox, 1962; McCracken and Dorn, 1964). 
Utilizing the generated values of DO deficit, the frequency distribution 
of probable values was computed and these are shown in Table 13, and 
plotted in Figures 14 through 19. The expected and most probable values 
of DO deficits for each of these cases are shown in Table 14. In order 
to evaluate and compare the results obtained by Monte Carlo techniques, 
predicted values of DO deficits were computed for each case using equa- 
tions postulated by Streeter and Phelps (1925), and Camp (1963). Average 
value of 0.173 per day for K^ and the value obtained from the regression 
equation, Eq, 31, for K 2 using the hydraulic characteristics reported 
in Table 11 were used in these formulae. These results are presented 
in Table 14, Percent error in predicting DO deficit was computed by 
multiplying 100 with the fraction obtained by dividing the absolute 
difference between actual and predicted values of DO deficit with the 
actual deficit. 



107 



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FIGURE 14. FREQUENCY DISTRIBUTION OF DO DEFICIT 
IN CASE STUDY NO. 1 



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DO Deficit, ai/1 

FIGURE 15. FREQUENCY DISTRIBUTION OF DO DEFICIT 
IN CASE STUDY NO. 2 



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FIGURE 16. FREQUENCY DISTRIBUTION OF DO DEFICIT 
IN CASE STUDY NO. 3 



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FIGURE 17. FREQUENCY DISTRIBUTION OF DO DEFICIT 
IN CASE STUDY NO. 4 



112 






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FIGURE 18. FREQUENCY DISTRIBUTION OF DO DEFICIT 
IN CASE STUDY NO. 5 



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DO Deficit, ag/1 



FIGURE 19. FREQUENCY DISTRIBUTION OF DO DEFICIT 
IN CASE STUDY NO. 6 



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114 



VII. DISCUSSION 

The significance and importance of considering the variations 
in K and K were brought out in Chapter III. In the carefully selected 
river reaches of the Tennessee Valley rivers, the atmospheric reaeration 
was found to vary significantly in every reach considered, even though 
each set of observations was made under constant river flow conditions. 
The observed range of values for K 2 was considerably more for shallow 
and high velocity flows than for deep and low velocity flows. As 
pointed out earlier, in one set of experiments in the Holston River, 
K 2 had a range of values from 0.10 to 1.18 per day with an arithmetic 
average of 0.63 per day. All the mathematical models developed for 
predicting the reaeration rate coefficient attempt to estimate only 
the average value for K 2 . Efforts so far made to evaluate the stream 
assimilative capacities fail to take into account the variations in 
the actual values of K 2 from the predicted values. These variations 
are considerable and lead to significant errors in predicting the DO 
response, if omitted. 

Attempts have been made to relate the reaeration coefficient 
to such hydraulic characteristics as velocity and depth of flow, energy 
slope, surface renewal rates, etc. These were not found to explain 
adequately the variations in the predicted values for K~. Inclusion of 
the variables, such as molecular diffusion, dispersion coefficient, 
surface tension, etc., did not significantly improve the predictions 
for reaeration coefficients (TVA, 1962). Variations in the values for 
reaeration coefficient may be attributed to the errors in estimation 



115 

of the parameters involved in the predictor equations, simplif i cat ion 
in the assumptions for developing the equations and other uncontrollable 
and unaccounted causes such as wind effects, turbulence, etc. As these 
factors are not quantifiable in a deterministic sense, the combined 
effects have to be expressed in terms of probability measure. For the 
Tennessee Valley rivers, the variations in reaeration coefficients have 
been found to be adequately defined by Gaussian distribution. 

Though the deoxygenation and reaeration coefficients were 
the only two parameters treated as random variables in this study, 
the probabilistic concepts should be extended to other factors such as 
photosynthesis, respiration, biological extraction of BOD, etc., which 
are subject to random fluctuations. Also the covariance of these factors 
needs to be given consideration. Attempts have been initiated at the 
University of Illinois to study the possible effects of different rates 
in deoxygenation on atmospheric reaeration capabilities in laboratory 
channels. Considerable additional research is needed in this area. 

Monte Carlo techniques applied to the problem of determining 
the DO response of streams receiving organic waste loads appear to 
yield satisfactory results. The results of the analysis of the Ohio 
River survey data presented in Table 14 indicate that there is con- 
siderable difference between the DO deficits actually observed and those 
predicted by the deterministic equations. Application of the Streeter- 
Phelps equation in which the effects of photosynthesis and respiration 
are omitted and in which the reaction coefficients are taken as constants 
in a river reach, results in prediction errors ranging from 0.5 to 33.5 
percent. Use of Camp's equation (1963) which accounts for an average 



116 

value of photosynthesis also results in prediction errors ranging from 
6.5 to 34,7 perdent. The use of an average value for photosynthesis 
in Camp's model irrespective of daytime or nighttime flow conditions 
is unrealistic since no photosynthesis can be expected to occur during 
nighttime. As the characteristics of the waste emanating even from a 
single source vary from time to time and the type and distribution of 
microflora stabilising the waste organic matter are bound to change 
in the natural environment, it is realistic to consider all the possible 
variations in parameters characterizing these changes instead of as- 
suming them to be time invariant. 

The percent error involved in predicting DO deficits using 
the probabilistic model range from 0.4 to 15.7 when the predictions 
are based on expected values and range from 0.0 to 11.5 when based on 
■est probable values. It is seen that under different conditions, the 
values of DO deficit predicted by the probabilistic model, particularly 
the most probable values, tend to be closer to the observed values all 
the time than the values predicted by the deterministic equations. 
The ooncept of most probable value is not unfamiliar in sanitary engi- 
neering practice, since the enumeration of coliform organisms in water 
samples involves this concept. 

The average value for the percent errors in DO deficits pre- 
dicted by the Streeter-Phelps equation for the six cases investigated 
in this study is 14.0 and the corresponding value for the Camp equation 
is 15.2. The average values in percent errors using the probabilistic 
model are 8,7 and 8.2 respectively for the expected value and most prob- 
able value cases. Thus it is seen that the use of most probable value 
predicted by the probabilistic model is likely to give more reliable 



117 



information on the DO response than the deterministic equations 
considered in this study. However the Streeter-Phelps model has pre- 
dicted the DO deficit within one percent error when the initial and 
final observations were carried out at night (Case Study No. 2) and 
the error in prediction using Camp's equation for this case is 6.5 
percent which is the best value obtained employing Camp's model. 

The errors inherent in the predictions using probabilistic 
model should be considered as being within practical limits, since 
several approximations and uncertainties creep in the evaluation of 
various factors which affect the DO-BOD relationship in streams. The 
estimation of time of flow between two sections in a river reach could 
at best be only approximate, so also the average values of depths and 
velocities of flow. Any errors in the assumption of oxygen saturation 
values at different temperatures are bound to be reflected in the final 
results of DO deficit predictions. Again there is the raging controversy 
about the adequacy of BOD bottle experiments to duplicate the oxidation 
process in the unrestricted natural environment. Sanitary engineers 
have to put up with this, being the only expedient method available at 
present to evaluate the biodegradability of organic matter. 

The use of the regression equation, Eq. 31, developed from 
the data collected in Tennessee Valley rivers, for computing mean 
values of K2 applicable to the Ohio River involves extrapolation to 
conditions other than those for which the equation was developed. The 
validity and usefulness of the equations developed by O'Connor and 



118 

Dobbins (1956) and Dobbins (1964) were not established beyond reasonable 
doubt. The discrepancies in observed and predicted values using these 
equations have been pointed out (Churchill et al., 1962; Thackston and 
Krenkel, 1965). The use of the regression equation developed for 
Tennessee Valley rivers, being the best available tool for estimating 
the reaeration coefficient independently, might be another source of 
error, since this involves extrapolation as indicated earlier. 

The strength of the proposed method lies in the fact that in 
the face of several ambiguities, it predicts a range of values for DO 
deficits with associated probabilities instead of a single value as in 
deterministic equations. Thus it enables one to quantify the uncer- 
tainties in terms of probability measure and to consider the probability 
of river DO being equal to or less than certain concentrations under 
given waste and stream flow conditions. 

Figure 20 shows the probable range of values for the oxygen 
sag profile in the Ohio River and the associated probabilities at a 
few selected sections for the initial conditions represented in Case 
Study No. 5. The histograms are drawn with class intervals of 0.25 
mg/1. Class intervals smaller than 0,25 mg/1 are considered unwarranted 
in view of the sensitivity of the DO determination methods. The pro- 
files predicted by Camp's and Streeter-Phelps ' equation are also shown 
in Figure 20, For short times of flow, the values of DO predicted by 
the two deterministic equations and the modal value of the probabilistic 
model are in close agreement within practical limits. As the time of 
flow increases, there is considerable deviation in the values predicted 
by these methods. The effect of the diurnal variations in the photosyn- 
thesis is seen to be significant during daytime flow conditions. 



119 



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s 



s 



w 

w 
fct 






5 

eg in 

z 

a* 

i-t H 

gen 

(A W 

33 



S « 

8 o 

Z f* 

<2 SB 

o2 o 



o 



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!/«■ '5iou«a oa 



120 

The probability distributions of DO at different sections of 
the river shown in Figure 20 pertain to the extent and intensity of 
exposure to the daylight. These factors depend on the time of initial 
observation at the upper end of the river reach considered. 

In the case of the probabilistic model, it is not possible to 
consider critical dissolved oxygen conditions as in deterministic 
models. The objective in waste water treatment systems design and 
operation will then be to minimize the probability of the dissolved 
oxygen level in the stream falling below the stipulated value. This 
could be evaluated knowing the probability distributions of the dis- 
solved oxygen concentrations at different sections of the river reaches 
for the known initial conditions. 

The writer became aware at a late stage of the possibility 
of employing alternative methods of finding theoretically the probability 
distribution function for the DO deficit, when the probability dis- 
tribution functions for Kj^ and K2 and their functional relationship 
with DO deficit are known. These alternatives are finding the dis- 
tribution of a function of one or more random variables using change of 
variable technique or the moment -generating function technique . The 
analytical technique will yield a more precise information than the 
Monte Carlo method. 

A need for taking into account on a rational basis the vari- 
ations in the parameters affecting the DO-BOD relationship had always 
been felt and this is at least partly met in this study. The diffi- 
culties one faces in predicting DO responses in a river system are best 



121 



summarized by quoting Kneese (1964): 

Much has been done to develop generalizations about 
the receiving water environment and to apply sci- 
entific principles to it. However precision is less 
and uncertainty greater in this area than in most fields 
of scientific and engineering forecasting. This em- 
phasizes the importance of empirical checking of 
forecasts made upon the basis of highly simplified 
principles. 



122 



VIII. ENGINEERING SIGNIFICANCE 

New methods are reported for evaluating the composite effects 
of combinations of waste water treatment, waste water flow regulation, 
stream flow regulation, optimal allocation of stream dissolved oxygen, 
etc. (Liebman and Lynn, 1966; Loucks and Lynn, 1966; Worley, 1963), for 
application in comprehensive programs to improve and maintain the 
quality of water in major water resources systems. Several new mathe- 
matical techniques useful in fusing engineering design, economic analy- 
sis and governmental planning have recently been developed to exploit 
the potentiality of the electronic computer. In these studies, the 
streams' self purification capacities and the allocation of stream dis- 
solved oxygen resources among various sources of pollution play an im- 
portant role and the reliability of the conclusions in these studies 
depends on the accuracies of dissolved oxygen predictions, to a smaller 
or larger extent. Thus the need for estimating the response of receiving 
streams to waste loads as accurately as possible becomes obvious. 

Kneese (1964) advocates the need for considering the proba- 
bilistic character of "damage costs" which include the optimal com- 
bmination of water treatment costs and physical damages due to the waste 
discharged upstream of the point of use. The mathematical expectation 
of damages associated with a particular level of stream flow is obtained 
by multiplying the probability of each flow by the corresponding cost 
(treatment plus damage). If, in addition to the probability of occurrence 
of different stream discharges, the variations in the basic velocity 
coefficients which define the streams* self purification capacities are 



123 



taken into account, a better estimate of the mathematical expectation 
of damages could be obtained* This in turn aids in management decisions 
aimed toward the best possible development of a water resources system. 
There is a growing realisation for the need of probabilistic 
oxygen standards in streams (Loucks, 1965; Thayer and Krutchkoff, 1966). 
In the earlier studies either the chance variations in stream flow 
(Loucks, 1965), or the variations in the states of BOD and 00 (Thayer 
and Krutchkoff, 1966) were considered. In these cases the velocity 
coefficients K, and K2 were treated as constants. Since the variations 
in Ki and K2 have been demonstrated to be significant, it is more realis 
tic to consider the variations in these coefficients within their 
practical ranges. The concepts developed in this work can be extended 
to include variations in stream flows, thus obtaining more general 
information on the probability distribution of dissolved oxygen concen- 
tration in the stream. In view of the uncertainties involved in the 
stream self purification process, it is more realistic to introduce 
arguments of probability and predict, not how the system will certainly 
and invariably perform, but how it will perform a certain fraction of 
the time. 



124 



IX. SUMMARY, CONCLUSIONS, AND SUGGESTIONS 
FOR FUTURE STUDY 



SUMMARY OF THE STUDY 

A digital computer model is used for defining the self 
purification process in the Ohio River-Cincinnati Pool reach. The 
model being a representation of the prototype Incorporating those 
features of the prototype deemed to be important for the purpose at 
hand, the operation of the model in any time Interval is character- 
ized by the inputs (BOD) in accordance with the parameters of the 
process, yielding a sequence of outputs (DO deficit). If the system 
is affected or perturbed by random components implicit in the input, 
any single simulation run yields only a specific solution related to 
one set of conditions of the system. To determine the general relation 
of the output to the input and variables may require a large number of 
simulation runs. Monte Carlo techniques are combined with simulation 
analysis to remedy the lack of generality implicit in simulation 
solutions. 

In applying Monte Carlo simulation techniques to the Ohio 
River-Cincinnati Pool reach, the mechanisms of BOD removal with con- 
comitant removal of oxygen due to bacterial oxidation of organic matter, 
oxygen addition due to photosynthesis and atmospheric reaeration and 
algal respiration are considered. Other known mechanisms affecting 
the DO-BOD relationship such as removal or addition of BOD due to 
deposition or scour, oxygen demand due to benthal deposits, etc., are 
considered insignificant in the oxygen balance studies for the particular 
case investigated. Oxygen addition due to photosynthesis is treated as a 



125 

periodic function having zero values at sunrise and sunset and a 
maximum value at mid-day. Only arithmetic average values for algal 
respiration and maximum rate of oxygen addition due to photosynthesis 
are considered. The reaction coefficients K^ and K~ are treated as 
random variables. A hypothetical case using Streeter -Phelps * formulation 
is used to establish the need for considering the variations in Kj and 

Two hundred values f»r K. and K2 are generated by Monte 
Carlo methods. Using these generated values of the reaction coef- 
ficients and average values for other parameters involved in the process, 
200 values for the dissolved oxygen deficit are generated, giving a 
probable range of values for the dissolved oxygen deficit with assoc- 
iated probabilities. The results of the dissolved oxygen deficits 
predicted by the probabilistic model are compared with the observed 
values. The predictions for dissolved oxygen deficits based on the 
most probable values are found to be in close agreement all the time 
with the observed values. 

To summarize the procedure involved in the use of Monte 
Carlo techniques for determining the self purification capacities of 
a stream under steady state flow conditions, the first step is to define 
all the mechanisms which affect BOD and DO, and to formulate a mathe- 
matical model describing the DO response. Probability distributions 
for the significant parameters which are found to vary considerably in 
the model are determined based on actual observations for these param- 
eters. The number of simulation runs required to predict the DO 
response under the known initial conditions is dictated by the precision 



126 



and the confidence with which it is to be predicted. The required 
number of values for the parameters which are found to vary with 
known probability distributions are obtained by suitable transformation 
techniques from generated random numbers. These generated values, 
which should conform to the probability distributions of the historical 
samples, are then used in the simulation model to obtain probable range 
of values for the 00 response with its associated probability 
distribution. 

CONCLUSIONS 

On the basis of this study, the following conclusions can be 
drawn: 

1. The variations in the values of deoxygenation and re- 
aeration coefficients within their practical ranges have significant 
effect in the prediction of the state of dissolved oxygen in streams 
receiving organic waste loads. Consequently there is little justifica- 
tion in considering only the average values for these coefficients 
instead of taking into account all the probable values in any given 
reach. 

2. The error in prediction of dissolved oxygen due to the 
variations in reaction coefficients increases with increase in 
temperature. 

3. The values for deoxygenation coefficient for river 
samples obtained downstream of all the major waste outfalls in Cin- 
cinnati, Ohio are found to vary at random. This is probably due to 
the random changes in the characteristics of the waste and the changes 



127 

in the type and distributions of microorganisms responsible for the 
stabilization of organic matter. The variations in deoxygenation co- 
efficient values are adequately represented by Gaussian distribution 
with a mean value of 0.173 per day, a standard deviation of 0.066 per 
day and a coefficient of variation of 38 percent. 

4, The variations in percent error in the prediction of re- 
aeration coefficient values for Tennessee Valley rivers using regression 
equation with mean depth and velocity of flow as independent variables 
follow normal distribution law with a mean of zero and a standard devi- 
ation of 36.8 percent. 

5. The most probable values for the dissolved oxygen deficits 
predicted by the probabilistic model using Monte Carlo simulation tech- 
niques are found to be better estimates than the values predicted by 
the conventional deterministic approaches. 

SUGGESTIONS FOR FUTURE STUDY 

Based on the results of this study and the understanding of 
the probabilistic process of the DO-BOD relationship in streams, the 
following suggestions for future research are proposed: 

1. The applicability of Monte Carlo techniques has to be 
verified for shallow streams where other mechanisms, in addition to 
the ones considered in this study, affecting the DO-BOD relationship 
are significant. 

2. The interdependence of the reaction coefficients K^ and 
Kj need further investigation. If they are not found completely in- 
dependent, as have been assumed in this study, a suitable joint density 
function has to be developed and used in the probabilistic model. 



128 



3* The variations in other parameters like oxygen addition 
due to photosynthesis, algal respiration, etc., need to be studied 
and accounted for in the probabilistic model, 

4. The concept of a simulation technique using Monte Carlo 
methods may be extended to include variable initial conditions arising 
from temporal variations in river flow, waste flow and waste strength. 

5. Research in the area of optimal design of waste treatment 
facilities, optimal allocation of stream dissolved oxygen, etc., should 
include the probabilistic variations in the parameters affecting 
streams.' waste assimilative capacities. 

6. The probabilistic model proposed In this study has to be 
solved analytically to determine the probability distribution function 
for the DO deficit employing either the change of variable technique 
or the moment -generating function technique. 



129 



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136 



APPENDIX A 
NOTATION 



137 



A} Coefficients in the linear regression equation. 

C a Constant defining the critical value of •<!• statistic in 
Kolmogorov-Smimov test for samples of size greater than 
35. 

C Dissolved oxygen concentration, mg/1, in stream at any time. 

C s Saturation DO concentration, mg/1, at any given temperature. 

c.d.f • Cumulative distribution function. 

D DO deficit, mg/1, at any given temperature and time of flow. 

D a Initial DO deficit, mg/1, in a river reach. 

E^ Oxygen demand, mg/(l)(day), due to benthal deposits. 

D Critical DO deficit, mg/1, in a stream. 

D n Coefficient of molecular diffusion, L 2 /T. 

(^(n) Critical value of 'd* statistic in Kolmogorov-Smirnov test 
for a sample size of n observations at a significance level 
of a. 

E(») Expected value of a function. 

ej Expected number of samples in category i in chi -square test. 

F(x) Cumulative frequency distribution. 

f Degree of freedom in chi -square test. 

f(x) Probability density function. 

g Number of population parameters estimated from the sample. 

H Mean depth, feet. 

Kj. Rate coefficient, day"*, defining BOD removal with con- 
comitant DO removal process (base e). 

K 2 Rate coefficient, day*"*, defining atmospheric reaeration 
process (base e). 

K3 Rate coefficient, day" 1 , defining addition or removal of 
BOD process due to scour or deposition. 

K5 Rate coefficient, day" 1 -, defining BOD removal due to biological 
extraction and accumulation. 



138 



K_ Rate coefficient, day"* 1 , defining overall removal rate 
of BOD in a river system. 

k Number of observations less than or equal to a given value x. 

L First stage ultimate BOD, mg/1, remaining to be satisfied. 

La Initial first stage ultimate BOD, mg/1, in a river reach. 

n Total number of observations made in a series of observations. 

nj Number of samples observed in category i of chi -square test. 

P{*] Probability of observing a value less than or equal to a 
stated value. 

P| Number of phases of length i in "turning points" test. 

P m Maximum rate of oxygen addition, mg/(l)(day), due to photo- 
synthesis. 

P t t Rate of oxygen addition, mg/(l)(day), at any given time. 

p Total number of peaks in "turning points" test and period 
of oscillation in a periodic sine function. 

p.d.f. Probability density function. 

R Average rate of algal respiration, sag/ (1) (day), and total 
number of runs in the "runs up and down" test. 

Rjj Number of runs of length k or more in the "runs up and down" 
test. 

Ro,i > 2,.,.,p Multiple correlation coefficient. 

r Total number of categories in chi -square test. 

z*l Number of runs of length i in the "runs up and down" test. 

S Slope of river channel. 

2 
Sg Sum of squared errors in regression analysis. 

s n(x) Observed cumulative step function of a sample of observations. 

T River temperature, degrees centigrade. 

t Time of flow, days. 



139 



t' Time of flow, in days, of a mass of water since sunrise 
after that mass of water has entered the head end of a 
river reach considered, 

t_ Flow through time, days, for critical section in a riveri 

V Mean velocity of flow, ft/sec, in a river reach. 

V(«) Variance of a function. 

X Random variable. 

x Any particular value a random variable X assumes. 

Z i Independent variables in regression analysis. 

Z Q Dependent variable value predicted in linear regression 
equation. 

Zq' Actual value of dependent variable. 

a Significance level. 

3 Error level in predicted values. 

A Incremental value in the variables. 

Temperature coefficient defining deoxygenation rates at 
different temperatures. 

$ Temperature coefficient defining reaeration rates at 
different temperatures. 

a Variance in population. 

X 2 Chi -square statistic. 



140 



APPENDIX B 

.C T J DIAGRAM FOR OXYGEN SAG COMPUTATIONS; 
SENSITIVITY ANALYSIS 



ۥ$ 



Read temperatures 1 to 30 degrees centigrade and the corres- 
ponding DO saturation value*. 



Read the values of K,, K 2 , D,, L,, river water temperature, 
maximum values of K, (DMK1) and K 2 (AMK2) and Incremental v< 
In «! OBIUKU and K 2 (AINK2). 



Find the DO saturation value corresponding to the river tempera- 
ture. Compute Initial percentage DO saturation and write output 
of river water temperature. D T . L«. Cb and Initial DO. 



141 



Set PRESER - DEOXK1 
I - 1 
J - 1 




D e - L a exp (-K^) 




- r%~ [exp(-K 1 t)-exp(.K 2 t) 
K 2"*l 

+ D a exp(-K 2 t)] 



Call system 
error 
































^--^^ riTN 











NK1 



IK, 



AMK2 . K 2 

ainkV 



+ 1.0 



IK, - DMK1 ' K W m 



DINK1 



(K, - K.)D 



arc - 1.0 - -jqr; 



l'"a 



t - 0.0 

fi - D. 




10 2 
~^~ ln(;r^ARG) 
K 2 -K 1 14 



K l L a 
K 2 exp(-K 1 t e ) 





D-D exp(-K 1 t)+ KjL a t expC-Kjt) 







- 


^C 


-~D~~- 

[0 


+ 


D « - s 











DO - C s - D 
Percent saturation 
t - t + 1.0 



C. - D 



x 100 



t 


^--^"■~ 












oj 


' 


Write output Kj, 
and critical DO. 


K 2 , 


t c , 


D c 









J - J + 1 



Write output sero to five days, the corresponding 
deficits, dissolved oxygen concentrations, and per- 
cent saturations for a particular combination of 
values for K t and K 2 . 




142 



APPENDIX C 

FLOW DIAGRAM FOR SOLVING PROBABILISTIC MODEL 
FOR DC-BOD RELATIONSHIP 



143 



G> 



(Start) 



Read input data; !)„, L a , V, 1?, T, R, time of initial observation 
and other relevant parameters. 



Generate a sequence of uniform (0,1) random numbers and evaluate 
the corresponding Kj values. Correct the unreasonably high or 
low values of K*. 



Perform Kolmogorov-Smirnov test on the generated values of K., 



Is the generated sample significantly different from the 
historical sample? 



Yes 



No 



Compute the value of trend component of K using regression 
equation. 



Generate a sequence of uniform (0,1) random numbers and evalu- 
ate the cor resp aiding random components of K 9 values. Correct 
the unreasonably high or low values for the random components 
of K«. 



Perform Kolmogorov-Smirnov test on the generated values for the 
random components of K«, 



Is the generated sample significantly different from the as- 
sumed distribution? 



Yes 



Develop random samples of K2 values by adding the trend com- 
ponent to the random components of K 9 , 



Set up necessary initial conditions for solving the differential 
equations using the sub-routine for fourth order F.unge-Kutta 
method. 



<D 



Use a set of generated values for K. and K„, apply tempera- 
ture corrections to them. Compute oxygen deficit for the 
known initial conditions using appropriate values for other 
parameters in the process. 



£ 



144 



Has the last set of generated values for K, and K^ been used? r.o-h 



i 



Yes 



Go to c 



Arrange the computed values for the DO deficits in increasing 
order and print the results. 



Have all input data been processed? 



r;o 



♦ Go to B 



Yes 




End 



145 



APPENDIX D 

BOD PROGRESSION DATA FOR 
THE OHIO RIVER SAMPLES (1957 SURVEY) 



146 





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155 



APPENDIX E 

HYDRAULIC CHARACTERISTICS OF REACHES IN 
TENNESSEE VALLEY RIVERS USED IN 
MULTIPLE CORRELATION ANALYSIS 



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